In the mathematical fields of topology and K-theory, the Serre–Swan theorem, also called Swan's theorem, relates the geometric notion of vector bundles to the algebraic concept of projective modules and gives rise to a common intuition throughout mathematics: "projective modules over commutative rings are like vector bundles on compact spaces".
The complementary variant stated by Richard Swan in 1962 is more analytic, and concerns (real, complex, or quaternionic) vector bundles on a smooth manifold or Hausdorff space.
Swan's theorem states that this module is finitely generated and projective over C∞(M).
This can be done by, for instance, exhibiting sections s1...sk with the property that for each point p, {si(p)} span the fiber over p. When M is connected, 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 × 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).
but is not 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 of smooth vector bundles over M to the category of finitely generated, projective C∞(M)-modules is full, faithful, and essentially surjective.
Suppose X is a compact Hausdorff space, and C(X) 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(X).
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(X) be the category of complex vector bundles over X, and let ProjMod(C(X)) be the category of finitely generated projective modules over the C*-algebra C(X).
There is a functor Γ : Vec(X) → ProjMod(C(X)) which sends each complex vector bundle E over X to the C(X)-module Γ(X, E) of sections.
and it follows that giving the map which respects the module structure (Várilly, 97).
Swan's theorem asserts that the functor Γ is an equivalence of categories.
The analogous result in algebraic geometry, due to Serre (1955, §50) applies to vector bundles in the category of affine varieties.
is the sheaf of germs of a finite-dimensional vector bundle if and only if
This article incorporates material from Serre-Swan theorem on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.