Suppose M is a compact smooth manifold, and a V is a smooth vector bundle over M. The space of smooth sections of V is then a module over C∞(M) (the commutative algebra of smooth real-valued functions on M).
The converse is also true: every finitely generated projective module over C∞(M) arises in this way from some smooth vector bundle on M. Such a module can be viewed as a smooth function f on M with values in the n &times; n idempotent matrices for some n.