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Serre–Swan theorem
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 two precise formulations of the theorems differ somewhat. The original theorem, as stated by Jean-Pierre Serre in 1955, is more algebraic in nature, and concerns vector bundles on an algebraic variety over an algebraically closed field (of any characteristic). 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.
== Differential geometry ==
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''). Swan's theorem states that this module is finitely generated and projective over C∞(''M''). In other words, every vector bundle is a direct summand of some trivial bundle: ''M'' × C''n'' for some ''n''. The theorem can be proved by constructing a bundle epimorphism from a trivial bundle ''M'' × C''n'' onto ''V''. This can be done by, for instance, exhibiting sections ''s''1...''s''''n'' with the property that for each point ''p'', span the fiber over ''p''.
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''). Therefore, the category of smooth vector bundles on ''M'' is equivalent to the category of finitely generated projective modules over C∞(''M''). Details may be found in . This equivalence is extended to the case of a noncompact manifold ''M'' (Giachetta et al. 2005).
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