Editing Serre–Swan theorem

{{short description|Relates the geometric vector bundles to algebraic projective modules}}
In the [[mathematics|mathematical]] fields of [[topology]] and [[K-theory]], the '''Serre–Swan theorem''', also called '''Swan's theorem''', relates the geometric notion of [[vector bundle]]s to the algebraic concept of [[projective module]]s and gives rise to a common intuition throughout [[mathematics]]: "projective modules over [[commutative ring]]s are like vector bundles on [[compact space]]s".

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 (algebra)|characteristic]]). The complementary variant stated by [[Richard Swan]] in 1962 is more analytic, and concerns ([[real number|real]], [[complex number|complex]], or [[quaternions|quaternionic]]) vector bundles on a [[smooth manifold]] or [[Hausdorff space]].

== 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}}.

== Topology ==
Suppose ''X'' is a compact [[Hausdorff space]], and C(''X'') is the ring of [[continuous function (topology)|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 number]]s.

In detail, let Vec(''X'') be the [[category theory|category]] of [[complex vector bundle]]s over ''X'', and let ProjMod(C(''X'')) be the category of [[finitely generated module|finitely generated]] projective modules over the [[Cstar algebra|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 [[section (fiber bundle)|sections]]. If <math>\tau : (E_1, \pi_1) \to (E_2, \pi_2)</math> is a [[Vector bundle#Vector bundle morphisms|morphism of vector bundles over ''X'']] then <math>\pi_2 \circ \tau = \pi_1</math> and it follows that

:<math>\forall s \in \Gamma(X, E_1) \quad \pi_2 \circ \tau \circ s = \pi_1 \circ s = \text{id}_X,</math>

giving the map 

:<math>\begin{cases} \Gamma \tau : \Gamma(X, E_1) \to \Gamma(X, E_2) \\ s \mapsto \tau \circ s \end{cases}</math>

which respects the module structure {{abbr|(Várilly, 97)|Várilly, Josef C. (1997), ''An introduction to noncommutative geometry'', eprint {{arXiv|physics/9709045v1}} [math-ph]: 6-7. Retrieved 4 June 2015.}}. Swan's theorem asserts that the functor Γ is an [[equivalence of categories]].

==Algebraic geometry==
The analogous result in [[algebraic geometry]], due to {{harvtxt|Serre|1955|loc=§50}} applies to vector bundles in the category of [[affine variety|affine varieties]].  Let ''X'' be an affine variety with structure sheaf <math>\mathcal{O}_X,</math> and <math>\mathcal{F}</math> a [[coherent sheaf]] of <math>\mathcal{O}_X</math> -modules on ''X''.  Then <math>\mathcal{F}</math> is the sheaf of germs of a finite-dimensional vector bundle if and only if <math>\Gamma(\mathcal{F}, X),</math> the space of sections of <math>\mathcal{F},</math> is a projective module over the commutative ring <math>A = \Gamma(\mathcal{O}_X, X).</math>

== References ==
*{{citation|first=Max|last=Karoubi|authorlink=Max Karoubi| title=K-theory: An introduction|publisher=Springer-Verlag|series=Grundlehren der mathematischen Wissenschaften|year=1978|isbn=978-0-387-08090-1}}
*{{citation|first=Palanivel|last=Manoharan|title=Generalized Swan's theorem and its application
|journal=[[Proceedings of the American Mathematical Society]]| volume=123|year=1995|pages=3219–3223|jstor=2160685|doi=10.1090/S0002-9939-1995-1264823-X|doi-access=free|issue=10|mr=1264823}}.
*{{citation|first=Jean-Pierre|last=Serre|authorlink=Jean-Pierre Serre|title=Faisceaux algébriques cohérents
|jstor=1969915|pages=197–278 
|journal=[[Annals of Mathematics]]| volume=61|year=1955|doi=10.2307/1969915|issue=2|mr=0068874}}.
*{{citation|title=Vector Bundles and Projective Modules|authorlink=Richard Swan|first=Richard G.|last=Swan|journal=[[Transactions of the American Mathematical Society]] |volume=105|year=1962|pages=264–277|jstor=1993627|doi=10.2307/1993627|issue=2|doi-access=free}}.
*{{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}}
*{{Citation | last = Giachetta | first = G. | last2 = Mangiarotti | first2 = L. | author3-link = Gennadi Sardanashvily|last3= Sardanashvily | first3 = Gennadi | year = 2005 | title = Geometric and Algebraic Topological Methods in Quantum Mechanics | publisher = World Scientific | isbn = 981-256-129-3
}}.

{{PlanetMath attribution|id=4066|title=Serre-Swan theorem}}

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[[Category:Commutative algebra]]
[[Category:Theorems in algebraic topology]]
[[Category:Differential topology]]
[[Category:K-theory]]