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 $\phi:L\oplus A^{n-1} \to\!\!\!\to I$. Assume that either (a) $k=1$ and $n\geq 3$ or (b) $k$ is arbitrary but $n\geq 4$ is even. Then $P$ has a unimodular element.