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# Serre-Swan 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.

Type of Math Object:

Theorem

Major Section:

Reference

## Mathematics Subject Classification

46L85*no label found*

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## Recent Activity

Jul 5

new correction: Error in proof of Proposition 2 by alex2907

Jun 24

new question: A good question by Ron Castillo

Jun 23

new question: A trascendental number. by Ron Castillo

Jun 19

new question: Banach lattice valued Bochner integrals by math ias

new correction: Error in proof of Proposition 2 by alex2907

Jun 24

new question: A good question by Ron Castillo

Jun 23

new question: A trascendental number. by Ron Castillo

Jun 19

new question: Banach lattice valued Bochner integrals by math ias