A celebrated result of Swan [1] states that, on a compact Hausdorff space $X$, the category of finite rank complex vector bundles is equivalent to the category of finitely generated projective $\mathcal{C}(X)$-modules, where $\mathcal{C}(X)$ is the ring of complex-valued continuous functions on $X$ (analogous statements are true over the reals and the quaternions (loc.cit.), but these will not concern us here). One can look upon this in the following ways:
interesting geometric objects and their morphisms-- viz. complex vector bundles of finite rank and vector bundle maps -- have a description by interesting algebraic objects and their morphisms viz. finitely generated projective modules and their morphisms;
just the other way round: interesting algebraic objects and their morphisms have a description by interesting geometric objects and their morphisms;
as a synthesis of these views one obtains that a topological invariant, the complex $K$-theory $K(X)$ of $X$, has both a geometric description by complex vector bundles and an algebraic description by projective modules.
Pondering a while upon these issues the following questions come to mind:
- The category of finite rank vector bundles is additive, but not abelian. This may lead one to consider linear fibrations over $X$ which are locally modelled on the kernels or cokernels of vector bundle maps of finite rank vector bundles; these have fibres which are vector spaces of possibly varying finite dimension. To which algebraic objects do these correspond?
- Are there further algebraic categories of $\mathcal{C}(X)$-modules with an interesting geometric description? In particular, do the finitely generated $\mathcal{C}(X)$-modules have an interesting geometric description?
- Does the topological $K$-homology $K_*(X)$ of $X$ have interesting geometrical and algebraic descriptions? The $K$-theory of finiteley generated $\mathcal{C}(X)$-modules appears to fail here, since pushing forward does not seem to work.
[1] Swan, R.G. -- Vector bundles and projective modules. Trans. Amer. Math. Soc. 105, (1962), 264–277.
