Existence of unimodular elements in a projective module

Abstract

Let $R$ be an affine algebra over an algebraically closed field of characteristic $0$ with dim$(R)=n$. Let $P$ be a projective $A=R[T_1,\cdots ,T_k]$-module of rank $n$ with determinant $L$. Suppose $I$ is an ideal of $A$ of height $n$ such that there are two surjections $\alpha :P\to \to I$ and $\varphi :L\oplus A^{n-1}\to \to I$. Assume that either (a) $k=1$ and $n\ge 3$ or (b) $k$ is arbitrary but $n\ge 4$ is even. Then $P$ has a unimodular element.