Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

The Serre-Swan theorem states (at least in one form) that the category of real vector bundles over a compact Hausdorff space $M$ is equivalent to the category of finitely generated projective modules over the ring $C(M)$ of continuous functions on $M$. The equivalence is provided by the functor $\Gamma$ sending a bundle to the totality of all its continuous sections.

Are there any classic applications or uses of this theorem? To me right now it seems like a pristine result to be admired from a distance, as I currently know of no actual use for it. I'd love to remedy that!

share|improve this question
    
Yeah, you're right; edited! –  Zach Conn Jun 29 '11 at 1:12
3  
Serre-Swan always struck me as more philosophically important than anything else. –  Qiaochu Yuan Jun 29 '11 at 3:12

3 Answers 3

up vote 10 down vote accepted

One application of this is that topological K-theory (i.e., the K-theory of the exact category of vector bundles on the space) is the same thing as the "algebraic" $K_0$ of the ring of continuous functions (i.e., the K-theory of the exact category of finitely generated projective modules over that ring). So topological K-theory is a "special case" of algebraic K-theory (though the tools for proving things like Bott periodicity are very different from those used in proving general results on exact category in algebraic K-theory).

By the way, here is the corresponding result in algebra:

Algebraic vector bundles over an affine scheme $\mathrm{Spec} A$ are the same as finitely generated projective $A$-modules (let's say $A$ is noetherian). So a module is projective if and only if it is locally free, in algebra language.

share|improve this answer

You could perhaps do worse than consulting $\S 6.4$ of my commutative algebra notes: "Applications of Swan's Theorem." (You could definitely do better: see below.)

The first application I give is to show that the ring of real-valued continuous functions on $[0,1]$ is a connected ring in which each finitely generated projective module is free but for which there is a nonfree infinitely generated projective module. As I admit myself in the notes, it is possible to prove this purely algebraically and I allude to another proof taken from one of Lam's books, but the topological approach is a nice one.

The second application is a big one: it exhibits stably free non-free modules over the ring $\mathbb{R}[x_0,\ldots,x_n]/(x_0^2+\ldots + x_n^2 - 1)$ of polynomial functions on the $n$-sphere when $n \neq 0,1,3,7$. This is done by reducing to the known behavior of tangent bundles to the $n$-sphere in the usual differential topological setting.

By the way, I got this second application directly from Swan's paper. There are other applications given there as well...

share|improve this answer
    
Re first application: You may remember this thread. Re second application: This blurb of Keith Conrad's is extremely nice. –  t.b. Jun 29 '11 at 8:16

In the theory of spectral triples, Connes's reconstruction theorem tells you that for a commutative spectral triple $(A,H,D)$ of metric dimension $p$, $A \cong C^\infty(X)$ for a (specific!) compact oriented $p$-manifold $X$. You can then use Serre--Swan to conclude that $H \cong L^2(X,E)$ for a Hermitian vector bundle $E \to X$ and that $D$ can be viewed as an essentially self-adjoint first-order differential operator on $E$. Very crudely speaking, you can view this as establishing a bridge between the $K$-homology of the manifold $X$ and the $K$-homology of the $C^\ast$-algebra $C(X)$ compatible with the identification $K^i(C(X)) \cong K_{-i}(X)$ of $K$-theories.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.