Suppose M is a smooth manifold, and E is a smooth vector bundle over M. Then Γ, the space of smooth sections of E, is a module over C∞. Swan's theorem states that this module is finitely generated and projective over C∞. In other words, every vector bundle is a direct summand of some trivial bundle: for some k. The theorem can be proved by constructing a bundle epimorphism from a trivial bundle This can be done by, for instance, exhibiting sectionss1...sk with the property that for each point p, span the fiber over p. When M is connected, the converse is also true: every finitely generated projective module over C∞ arises in this way from some smooth vector bundle on M. Such a module can be viewed as a smooth functionf on M with values in the n × n idempotent matrices for some n. The fiber of the corresponding vector bundle over x is then the range of f. If M is not connected, the converse does not hold unless one allows for vector bundles of non-constant rank. For example, if M is a zero-dimensional 2-point manifold, the module is finitely-generated and projective over but is not free, and so cannot correspond to the sections of any vector bundle over M. Another way of stating the above is that for any connected smooth manifold M, the section functorΓ from the category of smooth vector bundles over M to the category of finitely generated, projective C∞-modules is full, faithful, and essentially surjective. Therefore the category of smooth vector bundles on M is equivalent to the category of finitely generated, projective C∞-modules. Details may be found in.
Topology
Suppose X is a compact Hausdorff space, and C is the ring of continuous real-valued functions on X. Analogous to the result above, the category of real vector bundles on X is equivalent to the category of finitely generated projective modules over C. The same result holds if one replaces "real-valued" by "complex-valued" and "real vector bundle" by "complex vector bundle", but it does not hold if one replace the field by a totally disconnected field like the rational numbers. In detail, let Vec be the category of complex vector bundles over X, and let ProjMod be the category of finitely generated projective modules over the C*-algebra C. There is a functor Γ : Vec → ProjMod which sends each complex vector bundleE over X to the C-module Γ of sections. If is a morphism of vector bundles over X then and it follows that giving the map which respects the module structure. Swan's theorem asserts that the functor Γ is an equivalence of categories.
