A Criterion for Splitting of a Projective Module in Terms of Its Generic Sections

Abstract Let $R$ be a smooth affine domain of dimension $d\geq 3$ over $\overline{{\mathbb{F}}}_p$ with $p\neq 2$. Let $P$ be a projective $R$-module of rank $d-1$ with trivial determinant. We prove that $P$ splits off a free summand of rank 1 if and only if $P$ surjects onto a complete intersection ideal of height $d-1$.