Idea: Projective modules over commutative rings are like bundles on compact spaces.
Suppose _M_ is a smooth manifold (not necessarily compact), and _E_ is a [smooth vector bundle](https://en.wikipedia.org/wiki/Smooth_vector_bundle "Smooth vector bundle") over _M_. Then _Γ(E)_, the space of smooth sections "Section (fiber bundle)") of _E_, is a module over $\mathcal{C}^{\infty}(M)$ (the commutative algebra of smooth real-valued functions on _M_). Swan's theorem states that this module is [finitely generated](https://en.wikipedia.org/wiki/Finitely-generated_module "Finitely-generated module") and a projective module over $\mathcal{C}^{\infty}(M)$. In other words, every vector bundle is a direct summand of some trivial bundle: $M\times \mathbb{R}^k$ for some $k$. The theorem can be proved by constructing a bundle epimorphism from a trivial bundle $M\times \mathbb{R}^k\rightarrow E$. This can be done by, for instance, exhibiting sections $s_1,...,sk$ with the property that for each point $p$, $\{s_i(p)\}$ span the fiber over $p$.
When $M$ is [connected](https://en.wikipedia.org/wiki/Connected_space "Connected space"), the converse is also true: every [finitely generated projective module](https://en.wikipedia.org/wiki/Finitely_generated_projective_module "Finitely generated projective module") over $\mathcal{C}^{\infty}(M)$ arises in this way from some smooth vector bundle on $M$. By the way, remember that a vector bundle is the same as a locally free sheaf, so this justifies the idea that projective modules are locally free O-modules, which is tipically misundestood as "projective modules are stalkwise free modules (see locally free module)".
Extracted from https://en.wikipedia.org/wiki/Serre%E2%80%93Swan_theorem.
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Author of the notes: Antonio J. Pan-Collantes
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