Serre–Swan theorem

In the mathematical fields of topology and K-theory, Swan's theorem, also called the Serre–Swan 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".

Differential geometry

Suppose "M" is a C^∞-manifold, and a smooth vector bundle "V" is given on "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", {"s"_"i"("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 harv|Nestruev|2003.

Topology

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. Swan's theorem asserts that the functor Γ is an equivalence of categories.

References

*citation|first=Max|last=Karoubi|title=K-theory: An introduction|publisher=Springer-Verlag|series=Grundlehren der mathematischen Wissenschaften|year=1978|isbn=978-0387080901
*citation|first=P.|last=Manoharan|title=Generalized Swan's Theorem and its Application
journal=Proceedings of the American Mathematical Society|volume=123|number=10|year=1995|pages=3219-3223|url=http://www.jstor.org/stable/2160685.
*citation|first=Jean-Pierre|last=Serre|authorlink=Jean-Pierre Serre|title=Faisceaux Algebriques Coherents
url=http://www.jstor.org/stable/1969915|pages=197-278
journal=Annals of Mathematics|volume=61|number=2|year=1955.
*citation|title=Vector Bundles and Projective Modules|authorlink=Richard Swan|first=Richard G.|last=Swan|journal=Transactions of the American Mathematical Society|volume=105|number=2|year=1962|pages=264-277|url=http://www.jstor.org/stable/1993627.
*citation|first=Jet|last=Nestruev|title=Smooth manifolds and observables|publisher=Springer-Verlag|series=Graduate texts in mathematics|volume=220|year=2003|isbn=0-387-95543-7