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Serre-Swan theorem (Theorem)

Let $ X$ be a compact Hausdorff space. Let $ \mathord{\mathbf{Vec}}(X)$ be the category of complex vector bundles over $ X$. And, let $ \mathord{\mathbf{ProjMod}}(C(X))$ be the category of finitely generated projective modules over the $ C^*$-algebra $ C(X)$. There is a functor $ \Gamma\colon \mathord{\mathbf{Vec}}(X) \to \mathord{\mathbf{ProjMod}}(C(X))$ which sends each complex vector bundle $ E \to X$ to the $ C(X)$-module $ \Gamma(X,E)$ of continuous sections.

The functor $ \Gamma$ is an equivalence of categories.



"Serre-Swan theorem" is owned by mhale.
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Cross-references: equivalence of categories, sections, continuous, functor, finitely generated projective modules, vector bundles, complex, category, Hausdorff space, compact
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This is version 2 of Serre-Swan theorem, born on 2003-02-26, modified 2004-04-16.
Object id is 4066, canonical name is SerreSwanTheorem.
Accessed 2513 times total.

Classification:
AMS MSC: 46L85 (Functional analysis :: Selfadjoint operator algebras :: Noncommutative topology)
Pending Errata and Addenda
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