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.

单词	SerreSwanTheorem
释义	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.
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