Contents

Idea

The Serre-Swan theorem identifies suitable modules over an algebra of functions on some space with the modules of sections of vector bundles over that space and thereby identifies these modules with vector bundles themselves.

Together with theorems like Gelfand duality, the Serre-Swan theorem is a central part of the general duality between geometry and algebra. In particular it may serve to generalize the notion of vector bundle from standard geometry to more exotic forms of geometry, such as noncommutative geometry.

Statement

There are two different original theorems of the same intuitive spirit which are usually jointly called the Serre-Swan theorem, the first one is in algebraic geometry, the second in topology:

1) Serre’s theorem (Serre 55): let $R$ be a commmutative unital Noetherian ring (in particular, the coordinate ring of an affine variety over a field), then the category of finitely-generated projective $R$-modules is equivalent to the category of algebraic vector bundles (= locally free sheaves of structure sheaf-modules of constant finite rank) on $Spec R$.

2) Swan’s theorem (Swan 62): Given a Hausdorff compact space $X$, the category of finitely generated projective modules over the continuous-function algebra $C(X)$ is equivalent to the category of finite-rank vector bundles on $X$, where the equivalence is established by sending a vector bundle to the its module of continuous sections.

But there are also various variations of these theorems, for instance to differential geometry:

3) smooth Serre-Swan theorem (Nestruev 03, 11.33) For $X$ a smooth manifold with $\mathbb{R}$-algebra of smooth functions $C^\infty(X)$ there is an equivalence of categories between that of finite rank smooth vector bundles over $X$ and finitely generated projective modules over $C^\infty(X)$.

A general statement of the Serre-Swan theorems over ringed spaces is in (Morye).

If one drops the condition that the sheaf of modules over the structure sheaf of a ringed space is locally free, and allows it instad to be just locally presentable, then one arrives at the notion of quasicoherent sheaf of modules. Here the Serre-Swan theorem serves to clarify in which sense precisely these are generalizations of vector bundles.

The condition that the modules be projective can also naturally be relaxed. In higher geometry the Serre-Swan theorem becomes not only more general but also conceptually simpler: if instead of modules one considers chain complexes of modules ((∞,1)-modules) then under mild assumptions (see at projective resolution) every chain complex of modules is equivalent (quasi-isomorphic) to a chain complex of projective modules, and hence this condition in the statement of the traditional Serre-Swan theorem becomes automatic. Or in other words, the non-projective modules also do correspond to vector bundles, but to chain complexes of vector bundles (only that the chain homology of the complex is not itself a vector bundle again in this case). See at (∞,1)-vector bundle for more on this.

Applications

To K-theory

The Serre-Swan theorem serves to relate topological K-theory with algebraic K-theory. (…)

Isbell duality between algebra and geometry

$\phantom{A}$geometry$\phantom{A}$$\phantom{A}$category$\phantom{A}$$\phantom{A}$dual category$\phantom{A}$$\phantom{A}$algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$$\phantom{A}$$\overset{\text{Gelfand-Kolmogorov}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}$$\phantom{A}$$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$$\phantom{A}$$\overset{\text{Gelfand duality}}{\simeq} TopAlg^{op}_{C^\ast, comm}$$\phantom{A}$$\phantom{A}$comm. C-star-algebra$\phantom{A}$
$\phantom{A}$noncomm. topology$\phantom{A}$$\phantom{A}$$NCTopSpaces_{H,cpt}$$\phantom{A}$$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}$$\phantom{A}$$\phantom{A}$general C-star-algebra$\phantom{A}$
$\phantom{A}$algebraic geometry$\phantom{A}$$\phantom{A}$$\phantom{NC}Schemes_{Aff}$$\phantom{A}$$\phantom{A}$$\overset{\text{almost by def.}}{\hookrightarrow} \phantom{Top}Alg^{op}_{fin}$$\phantom{A}$$\phantom{A}$fin. gen.$\phantom{A}$
$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$noncomm. algebraic$\phantom{A}$
$\phantom{A}$geometry$\phantom{A}$
$\phantom{A}$$NCSchemes_{Aff}$$\phantom{A}$$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}$$\phantom{A}$$\phantom{A}$fin. gen.
$\phantom{A}$associative algebra$\phantom{A}$$\phantom{A}$
$\phantom{A}$differential geometry$\phantom{A}$$\phantom{A}$$SmoothManifolds$$\phantom{A}$$\phantom{A}$$\overset{\text{Milnor's exercise}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}$$\phantom{A}$$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$$\phantom{A}$$\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}$$\phantom{A}$$\phantom{A}$$\array{ \overset{\phantom{\text{Milnor's exercise}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }$$\phantom{A}$$\phantom{A}$supercommutative$\phantom{A}$
$\phantom{A}$superalgebra$\phantom{A}$
$\phantom{A}$formal higher$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$
$\phantom{A}$(super Lie theory)$\phantom{A}$
$\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}$$\phantom{A}\array{ \overset{ \phantom{A}\text{Lada-Markl}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}$$\phantom{A}$differential graded-commutative$\phantom{A}$
$\phantom{A}$superalgebra
$\phantom{A}$ (“FDAs”)

in physics:

$\phantom{A}$algebra$\phantom{A}$$\phantom{A}$geometry$\phantom{A}$
$\phantom{A}$Poisson algebra$\phantom{A}$$\phantom{A}$Poisson manifold$\phantom{A}$
$\phantom{A}$deformation quantization$\phantom{A}$$\phantom{A}$geometric quantization$\phantom{A}$
$\phantom{A}$algebra of observables$\phantom{A}$space of states$\phantom{A}$
$\phantom{A}$Heisenberg picture$\phantom{A}$Schrödinger picture$\phantom{A}$
$\phantom{A}$AQFT$\phantom{A}$$\phantom{A}$FQFT$\phantom{A}$
$\phantom{A}$higher algebra$\phantom{A}$$\phantom{A}$higher geometry$\phantom{A}$
$\phantom{A}$Poisson n-algebra$\phantom{A}$$\phantom{A}$n-plectic manifold$\phantom{A}$
$\phantom{A}$En-algebras$\phantom{A}$$\phantom{A}$higher symplectic geometry$\phantom{A}$
$\phantom{A}$BD-BV quantization$\phantom{A}$$\phantom{A}$higher geometric quantization$\phantom{A}$
$\phantom{A}$factorization algebra of observables$\phantom{A}$$\phantom{A}$extended quantum field theory$\phantom{A}$
$\phantom{A}$factorization homology$\phantom{A}$$\phantom{A}$cobordism representation$\phantom{A}$

The two original articles are

• Jean-Pierre Serre, Faisceaux algebriques coherents, Annals of Mathematics 61 (2): 197–278 (1955)

• Richard Swan, Vector bundles and projective modules, Trans. AMS 105 (2): 264–277 (1962)

A textbook account in the context of differential geometry is in

• Jet Nestruev, Smooth manifolds and observables, Graduate texts in mathematics, 220, Springer-Verlag, ISBN 0-387-95543-7 (2003)

A general account of Serre-Swan-type theorems over ringed spaces is in

A textbook account on the use of the theorem in K-theory is for instance

• Max Karoubi, $K$-theory. An introduction, Grundlehren der Mathematischen Wissenschaften, Band 226, Springer 1978. xviii+308 pp.