Editing Serre–Swan theorem (section)

== Differential geometry ==
Suppose ''M'' is a [[smooth manifold]] (not necessarily compact), and ''E'' is a [[smooth vector bundle]] over ''M''. Then ''Γ(E)'', the space of [[section (fiber bundle)|smooth sections]] of ''E'', is a [[module (mathematics)|module]] over C<sup>∞</sup>(''M'') (the commutative algebra of smooth real-valued functions on ''M''). Swan's theorem states that this module is [[finitely-generated module|finitely generated]] and [[projective module|projective]] over C<sup>∞</sup>(''M''). In other words, every vector bundle is a direct summand of some trivial bundle: <math>M \times \R^k</math> for some ''k''. The theorem can be proved by constructing a bundle epimorphism from a trivial bundle <math>M \times \R^k \to E.</math> This can be done by, for instance, exhibiting sections ''s''<sub>1</sub>...''s''<sub>''k''</sub> with the property that for each point ''p'', {''s''<sub>''i''</sub>(''p'')} span the fiber over ''p''. 

When ''M'' is [[connected space|connected]], the converse is also true: every [[finitely generated projective module]] over C<sup>∞</sup>(''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''. The fiber of the corresponding vector bundle over ''x'' is then the range of ''f''(''x''). If ''M'' is not connected, the converse does not hold unless one allows for vector bundles of non-constant rank (which means admitting manifolds of non-constant dimension). For example, if ''M'' is a zero-dimensional 2-point manifold, the module <math>\R\oplus 0</math> is finitely-generated and projective over <math>C^\infty(M)\cong\R\times\R</math> but is not [[free module|free]], and so cannot correspond to the sections of any (constant-rank) vector bundle over ''M'' (all of which are trivial).

Another way of stating the above is that for any connected smooth manifold ''M'', the section [[functor]] ''Γ'' from the [[category theory|category]] of smooth vector bundles over ''M'' to the category of finitely generated, projective C<sup>∞</sup>(''M'')-modules is [[full functor|full]], [[faithful functor|faithful]], and [[essentially surjective functor|essentially surjective]]. Therefore the category of smooth vector bundles on ''M'' is [[Equivalence of categories|equivalent]] to the category of finitely generated, projective C<sup>∞</sup>(''M'')-modules. Details may be found in {{harv|Nestruev|2003}}.