Serre-Swan theorem

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 over M. Then Γ(E), the space of smooth sections of E, is a module over ${\mathcal{C}}^{\mathrm{\infty }}(M)$ (the commutative algebra of smooth real-valued functions on M). Swan's theorem states that this module is finitely generated and a projective module over ${\mathcal{C}}^{\mathrm{\infty }}(M)$. 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 $M\times {\mathbb{R}}^k\to E$. 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, the converse is also true: every finitely generated projective module over ${\mathcal{C}}^{\mathrm{\infty }}(M)$ 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–Swan_theorem.