Serre-Swan theorem

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

The functor $\mathrm{\Gamma }$ is an equivalence of categories.

Title	Serre-Swan theorem
Canonical name	SerreSwanTheorem
Date of creation	2013-03-22 13:29:31
Last modified on	2013-03-22 13:29:31
Owner	mhale (572)
Last modified by	mhale (572)
Numerical id	5
Author	mhale (572)
Entry type	Theorem
Classification	msc 46L85