Critical-rank representatives of rank-zero classes

PDF p. 12: Proof of Theorem A, rank two representative; PDF p. 15: Proof of Theorem 6.7, first sentence; PDF p. 17: Proof of Theorem B, rank three representatives

Assume the Axiom of Choice and suitable universes. Let $X$ be a nonempty affine scheme of finite type over a field, of dimension $d=2$ or $3$. Every class $\alpha$ of pointwise rank zero in $K_0(X)={\pi }_{0}K(X)$ can be written $\alpha =[E]-[{\mathcal{O}}_X^d]$ for a vector bundle $E$ of constant rank $d$. Moreover, any vector bundle $V$ of constant rank $r\ge d$ is isomorphic to $E^{\mathrm{\prime }}\oplus {\mathcal{O}}_X^{r-d}$ for a vector bundle $E^{\mathrm{\prime }}$ of rank $d$.

A vector bundle is a finite locally free ${\mathcal{O}}_X$-module; its rank at a point is the number of copies of the structure sheaf in a local trivialization there. Constant rank means the same rank at every point. The dimension is the supremum of the lengths of strict chains of irreducible closed subsets. The spectrum $K(X)$ is the nonconnective K-theory of perfect complexes in the supplied interface. The class of a vector bundle in $K_0(X)$ is the class of the complex concentrated in degree zero. Pointwise rank on $K_0(X)$ is transported through the affine K-zero isomorphism to the rank homomorphism on the vector-bundle Grothendieck group: a difference $[B]-[C]$ has rank ${\mathrm{rank}}_{x}B-{\mathrm{rank}}_{x}C$ at $x$.