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

TL;DR

This paper proves that under certain conditions, a projective module over a polynomial ring has a unimodular element, extending known results to a broader class of rings and modules.

Contribution

It establishes a new criterion for the existence of unimodular elements in projective modules over polynomial rings, generalizing previous results by Mandal.

Findings

If P/TP and P_f contain unimodular elements, then P has a unimodular element.

The result applies to Noetherian rings of dimension n with projective modules of rank n.

The theorem extends the analogue of Mandal's result to polynomial extensions.

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.