A module $P$ is projective if and only if there is another module $Q$ such that the [direct sum](https://en.wikipedia.org/wiki/Direct_sum_of_modules "Direct sum of modules") of $P$ and $Q$ is a free module.
In mathematics, particularly in algebra, the class of projective modules enlarges the class of free modules (that is, modules with basis vectors) over a ring, by keeping some of the main properties of free modules. There are various equivalent characterizations (see wikipedia.
Every free module is a projective module, but the converse fails to hold over some rings, such as Dedekind rings that are not principal ideal domains.
A typical example of projective module is the set of sections of a vector bundle. Indeed, a $\mathcal{C}^{\infty}(M)$-module $P$ is projective and finitely generated if and only if is isomorphic to $\Gamma(M,E)$ for some vector bundle $E\rightarrow M$ ([Swan _1962], theorem 2).
See Serre-Swan theorem.
That is, projective $\mathcal{C}^{\infty}(M)$-modules are locally free O-modules.
________________________________________
________________________________________
________________________________________
Author of the notes: Antonio J. Pan-Collantes
INDEX: