Relation between algebraic and topological K-theories

Abstract

The natural relationship between algebraic and topological K-theories was established by Swan. Let X be a compact Hausdorff space and let k(X) be the ring of continuous functions on X with values in k = ℝ or C. If ξ is a real or complex vector bundle over X, the group Γ(ξ) of global sections can be viewed as a k(X)-module. Swan’s result says that the functor Γ establishes an equivalence between the category of vector bundles over X and the category of finitely generated projective k(X)-modules. In particular we have K ₀ (k(X)) = K _k ⁰ (X).