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

Abstract

If $n\equiv 0,1mod4$, 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 $char(k)\ne 2$ 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 $d\ge 4$ over algebraically closed fields of characteristic $\ne 2$. 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 $char(k)\ne 2,3$ is cancellative.

arXiv comment26 pages, to appear in JPAA