Swan's theorem
Encyclopedia
In the 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 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
.
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 × Cn for some n. The theorem can be proved by constructing a bundle epimorphism from a trivial bundle M × Cn onto V. This can be done by, for instance, exhibiting sections s1...sn with the property that for each point p, {si(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).
, 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 number
s.
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
.
, due to applies to vector bundles in the category of affine varieties. Let X be an affine variety with structure sheaf OX, and F a coherent sheaf
of OX-modules on X. Then F is the sheaf of germs of a finite-dimensional vector bundle if and only if the space of sections of F, Γ(F,X), is a projective module over the commutative ring A = Γ(OX,X).
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
fields of topology
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
and K-theory
K-theory
In mathematics, K-theory originated as the study of a ring generated by vector bundles over a topological space or scheme. In algebraic topology, it is an extraordinary cohomology theory known as topological K-theory. In algebra and algebraic geometry, it is referred to as algebraic K-theory. It...
, the Serre–Swan theorem, also called Swan's theorem, relates the geometric notion of vector bundle
Vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
s to the algebraic concept of projective module
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...
s and gives rise to a common intuition throughout mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
: "projective modules over commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
s 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
Jean-Pierre Serre
Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...
in 1955, is more algebraic in nature, and concerns vector bundles on an algebraic variety
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
over an algebraically closed field
Algebraically closed field
In mathematics, a field F is said to be algebraically closed if every polynomial with one variable of degree at least 1, with coefficients in F, has a root in F.-Examples:...
(of any characteristic
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
). The complementary variant stated by Richard Swan
Richard Swan
Richard Gordon Swan is an American mathematician who is best known for Swan's theorem. His work has mainly been in the area of algebraic K-theory.-External links:**...
in 1962 is more analytic, and concerns (real, complex, or quaternionic) vector bundles on a smooth manifold or Hausdorff space
Hausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
.
Differential geometry
Suppose M is a compact smooth manifold, and a V is a smooth vector bundleVector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X : to every point x of the space X we associate a vector space V in such a way that these vector spaces fit together...
over M. The space of smooth sections
Section (fiber bundle)
In the mathematical field of topology, a section of a fiber bundle π is a continuous right inverse of the function π...
of V is then a module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
over C∞(M) (the commutative algebra of smooth real-valued functions on M). Swan's theorem states that this module is finitely generated
Finitely-generated module
In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated R-module also may be called a finite R-module or finite over R....
and projective
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...
over C∞(M). In other words, every vector bundle is a direct summand of some trivial bundle: M × Cn for some n. The theorem can be proved by constructing a bundle epimorphism from a trivial bundle M × Cn onto V. This can be done by, for instance, exhibiting sections s1...sn with the property that for each point p, {si(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
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
of smooth vector bundles on M is equivalent
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
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).
Topology
Suppose X is a compact Hausdorff spaceHausdorff space
In topology and related branches of mathematics, a Hausdorff space, separated space or T2 space is a topological space in which distinct points have disjoint neighbourhoods. Of the many separation axioms that can be imposed on a topological space, the "Hausdorff condition" is the most frequently...
, 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 number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s.
In detail, let Vec(X) be the category
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
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
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
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
Section (fiber bundle)
In the mathematical field of topology, a section of a fiber bundle π is a continuous right inverse of the function π...
. Swan's theorem asserts that the functor Γ is an equivalence of categories
Equivalence of categories
In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics...
.
Algebraic geometry
The analogous result in algebraic geometryAlgebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
, due to applies to vector bundles in the category of affine varieties. Let X be an affine variety with structure sheaf OX, and F a coherent sheaf
Coherent sheaf
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a specific class of sheaves having particularly manageable properties closely linked to the geometrical properties of the underlying space. The definition of coherent sheaves is made with...
of OX-modules on X. Then F is the sheaf of germs of a finite-dimensional vector bundle if and only if the space of sections of F, Γ(F,X), is a projective module over the commutative ring A = Γ(OX,X).