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We study the cancellation property of projective modules of rank $2$ with atrivial determinant over Noetherian rings of dimension $\leq 4$. If $R$ is asmooth 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 ofrank $2$ are free if and only if a Hermitian $K$-theory group $\tilde{V}_{SL}(R)$ is trivial.

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