论文标题
取消对矢量捆的排名$ 3 $的零件班级$ 3 $
Cancellation of vector bundles of rank $3$ with trivial Chern classes on smooth affine fourfolds
论文作者
论文摘要
If $n \equiv 0,1~mod~4$, we prove a sum formula $V_{θ_{0}} (a_{0},a_{R}^{n}) = n \cdot V_{θ_{0}} (a_{0},a_{R})$ for the generalized Vaserstein symbol whenever $R$ is a smooth affine带有$ char（k）\ neq 2 $的完美字段$ k $的代数$ -1 \ in {k^{\ times}}}}^{2} $。这使我们能够通过基本矩阵对fasel-rao-swan的结果概括，而不是尺寸的正常仿射代数$ d \ geq 4 $，而不是代数封闭的特征$ \ neq 2 $。结果，我们证明，在平稳的仿射代数上，任何排名$ 3 $的投影模块都在尺寸的平滑仿射代数上$ 4 $，而代数封闭的字段$ k $，$ char（k）\ neq 2,3 $是取消的。
If $n \equiv 0,1~mod~4$, we prove a sum formula $V_{θ_{0}} (a_{0},a_{R}^{n}) = n \cdot V_{θ_{0}} (a_{0},a_{R})$ 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.
