Rank-three representatives with two zero Chern classes

PDF p. 15: First sentence of Theorem 6.7 proof

Assume the Axiom of Choice and the site and universe size bounds required by the completed filtration and Chern comparisons. Let $k$ be an algebraically closed field and let $X$ be a normal affine $k$-scheme of finite type and dimension three. Every $\alpha \in F_{\mathrm{m}\mathrm{o}\mathrm{t}}^3{K}_0(X)$ can be written $\alpha =[P]-[{\mathcal{O}}_X^3]$ for a rank-three vector bundle $P$ with $c_1(P)=0$, $c_2(P)=0$, and $\det P\cong {\mathcal{O}}_X$.

A vector bundle is a finite locally free ${\mathcal{O}}_X$-module, and rank three means that it is locally isomorphic to ${\mathcal{O}}_X^3$. Its determinant is the line bundle ${\bigwedge }^{3}P$. Normal means that every local ring is an integrally closed domain; finite type means that the affine coordinate ring is a finitely generated $k$-algebra. Dimension is the supremum of lengths of strict chains of irreducible closed subsets. Here $K_0(X)={\pi }_{0}K(X)$ for the specified nonconnective K-theory spectrum of perfect complexes, and $[P]$ denotes the class of $P$ concentrated in degree zero. The subgroup $F_{\mathrm{m}\mathrm{o}\mathrm{t}}^i{K}_0(X)$ is the image of the zeroth homotopy group of the $i$th motivic filtration stage in $K_0(X)$. The bundle Chern classes $c_i(P)$ lie in $H_{\mathrm{m}\mathrm{o}\mathrm{t}}^{2i,i}(X,\mathbb{Z})$ and are the classes defined by the supplied motivic projective-bundle relation. Choice asserts that every family of nonempty sets has a choice function; the size bounds fix the sites and ambient universes of the cited comparisons.