Unimodular elements in projective modules and an analogue of a result of Mandal

Abstract

(1) Let $R$ be a commutative Noetherian ring of dimension $n$ and $P$ a projective $R[X_1,\ldots ,X_m]$-module of rank $n$. In this paper, we associate an obstruction for $P$ to split off a free summand of rank one. (2) Let $R$ be a local ring and $R[X]\subset A\subset R[X,X^{-1}]$. Let $P$ and $Q$ be two projective $A$-modules with $\text {rank}(Q)\lt \text {rank}(P)$. If $Q_f$ is a direct summand of $P_f$ for some special monic polynomial $f\in R[X]$, then $Q$ is also a direct summand of $P$.