Affine bundle K zero is the projective split group

PDF p. 12: Bundle classes in spectral K zero

Assume the Axiom of Choice. For a commutative unital ring $A$ and $X=\mathrm{Spec}A$, associated sheaf induces an isomorphism from the split Grothendieck group of finitely generated projective $A$-modules to the vector-bundle Grothendieck group of $X$.

A finitely generated projective module is a finitely generated module with the lifting property through surjective module maps. A vector bundle is a locally finite free ${\mathcal{O}}_X$-module. Its Grothendieck group is generated by isomorphism classes $[E]$, with $[E]=[E^{\mathrm{\prime }}]+[E^{\mathrm{\prime }\mathrm{\prime }}]$ for every short exact sequence $0\to E^{\mathrm{\prime }}\to E\to E^{\mathrm{\prime }\mathrm{\prime }}\to 0$ of vector bundles. The split group has only the corresponding direct-sum relations.