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!

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    $\begingroup$ Serre-Swan always struck me as more philosophically important than anything else. $\endgroup$ – Qiaochu Yuan Jun 29 '11 at 3:12

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.

  • $\begingroup$ The result about algebraic vector bundles holds in full generality, not needing that $A$ is Noetherian. See for instance Theorem 7.22 in Pete Clark's notes. $\endgroup$ – Ingo Blechschmidt Mar 10 '15 at 10:00
  • $\begingroup$ Very late comment, but it seems worth mentioning that the higher topological $K$-groups of $X$ do not coincide with the higher algebraic $K$-groups of $C(X)$ (they do coincide with the higher operator algebraic $K$-groups). So "special case" is only true for $K_0$. $\endgroup$ – MaoWao Nov 28 '18 at 12:59

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...


One nice application, which took me a while to understand, (and Wikipedia was of great help!) is the beautiful "bridge" between Algebra and Differential Geometry, that the bundle of frames is trivial iff it has a (global, obviously) section. (this means in this case a basis of each tangent space, globally and of course differentiably defined). Serre-Swan theorem measures in this case how far a bunlde is from being isomorphic to a product, and at the same time, how far its space of sections is far from being free (over the ring of differentiable functions on the manifold)


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