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

TL;DR

This paper proves that rank 3 projective modules with trivial Chern classes over certain smooth affine fourfolds are cancellative, extending previous results and employing a generalized Vaserstein symbol sum formula.

Contribution

It introduces a sum formula for the generalized Vaserstein symbol applicable to smooth affine algebras over perfect fields, leading to new cancellation results for rank 3 modules.

Findings

Proves a sum formula for the generalized Vaserstein symbol when n ≡ 0,1 mod 4.

Extends Fasel-Rao-Swan results on unimodular row transformations.

Shows all rank 3 projective modules with trivial Chern classes over certain fourfolds are cancellative.

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.