Unimodular Elements in Projective Modules and an analogue of a result of Mandal

Abstract

Let $R$ be a Noetherian commutative ring of dimension $n$, $A=R[X_1,\cdots,X_m]$ be a polynomial ring over $R$ and $P$ be a projective $A[T]$-module of rank $n$. Assume that $P/TP$ and $P_f$ both contain a unimodular element for some monic polynomial $f(T)\in A[T]$. Then $P$ has a unimodular element.