Projective module

A module $P$ is projective if and only if there is another module $Q$ such that the direct sum 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 ${\mathcal{C}}^{\mathrm{\infty }}(M)$-module $P$ is projective and finitely generated if and only if is isomorphic to $\mathrm{\Gamma }(M,E)$ for some vector bundle $E\to M$ ([Swan _1962], theorem 2).
See Serre-Swan theorem.

That is, projective ${\mathcal{C}}^{\mathrm{\infty }}(M)$-modules are locally free O-modules.