The cancellation of projective modules of rank 2 with a trivial determinant

Abstract

We study the cancellation property of projective modules of rank $2$ with a trivial determinant over Noetherian rings of dimension $\le 4$. If $R$ is a smooth affine algebra of dimension $4$ over an algebraically closed field $k$ such that $6\in k^{\times }$, then we prove that stably free $R$-modules of rank $2$ are free if and only if a Hermitian $K$-theory group ${\tilde{V}}_{SL}\left(R\right)$ is trivial.