Serre–Swan theorem

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.


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.


*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=
*citation|first=Jean-Pierre|last=Serre|authorlink=Jean-Pierre Serre|title=Faisceaux Algebriques Coherents
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=
*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

Wikimedia Foundation. 2010.

См. также в других словарях:

  • Richard Swan — Richard Gordon Swan (* 21. Dezember 1933 in New York City) ist ein US amerikanischer Mathematiker, der sich mit Algebra (unter anderem mit algebraischer K Theorie) beschäftigt. Swan war 1952 Putnam Fellow und studierte an der Princeton University …   Deutsch Wikipedia

  • List of mathematics articles (S) — NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… …   Wikipedia

  • Nakayama lemma — In mathematics, more specifically modern algebra and commutative algebra, Nakayama s lemma also known as the Krull–Azumaya theorem[1] governs the interaction between the Jacobson radical of a ring (typically a commutative ring) and its finitely… …   Wikipedia

  • Exterior algebra — In mathematics, the exterior product or wedge product of vectors is an algebraic construction generalizing certain features of the cross product to higher dimensions. Like the cross product, and the scalar triple product, the exterior product of… …   Wikipedia

  • K-Theorie — Das mathematische Teilgebiet der K Theorie beschäftigt sich mit dem Studium von Vektorbündeln auf topologischen Räumen (topologische K Theorie) oder Ringen bzw. Schemata (algebraische K Theorie). (Der Name K Theorie wurde von Alexander… …   Deutsch Wikipedia

  • Sheaf (mathematics) — This article is about sheaves on topological spaces. For sheaves on a site see Grothendieck topology and Topos. In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space.… …   Wikipedia

  • Scientific phenomena named after people — This is a list of scientific phenomena and concepts named after people (eponymous phenomena). For other lists of eponyms, see eponym. NOTOC A* Abderhalden ninhydrin reaction Emil Abderhalden * Abney effect, Abney s law of additivity William de… …   Wikipedia

  • Séminaire Nicolas Bourbaki (1960–1969) — Continuation of the Séminaire Nicolas Bourbaki programme, for the 1960s.1960/61 series*205 Adrien Douady, Plongements de sphères, d après Mazur et Brown (embeddings of spheres) *206 Roger Godement, Groupes linéaires algébriques sur un corps… …   Wikipedia

  • Projective module — In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module (that is, a module with basis vectors). Various equivalent… …   Wikipedia

  • List of mathematics articles (B) — NOTOC B B spline B* algebra B* search algorithm B,C,K,W system BA model Ba space Babuška Lax Milgram theorem Baby Monster group Baby step giant step Babylonian mathematics Babylonian numerals Bach tensor Bach s algorithm Bachmann–Howard ordinal… …   Wikipedia

Поделиться ссылкой на выделенное

Прямая ссылка:
Нажмите правой клавишей мыши и выберите «Копировать ссылку»