Cancellation of vector bundles of rank $3$ with trivial Chern classes on smooth affine fourfolds

Abstract

If $n \equiv 0,1~mod~4$, we prove a sum formula $V_{\theta_{0}} (a_{0},a_{R}^{n}) = n \cdot V_{\theta_{0}} (a_{0},a_{R})$ for the generalized Vaserstein symbol whenever $R$ is a smooth affine algebra over a perfect field $k$ with $char(k) \neq 2$ such that $-1 \in {k^{\times}}^{2}$. This enables us to generalize a result of Fasel-Rao-Swan on transformations of unimodular rows via elementary matrices over normal affine algebras of dimension $d \geq 4$ over algebraically closed fields of characteristic $\neq 2$. As a consequence, we prove that any projective module of rank $3$ with trivial Chern classes over a smooth affine algebra of dimension $4$ over an algebraically closed field $k$ with $char(k) \neq 2,3$ is cancellative.