A Few Elements of K-Theory

Abstract

We have seen in the previous Section that the algebraic substitutes for bundles (of a particular kind, at least) are projective modules of finite type over an algebra $\mathcal{A}$. The (algebraic) K-theory of $\mathcal{A}$ is the natural framework for the analogue of bundle invariants. Indeed, both the notions of isomorphism and of stable isomorphism have a meaning in the context of finite projective (right) modules. The group $K_0 \left( \mathcal{A} \right)$ will be the group of (stable) isomorphism classes of such modules.