# Existence of unimodular elements in a projective module

## Article Type

Research Article

## Publication Title

Journal of Pure and Applied Algebra

## Abstract

(1) 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[T1,⋯,Tk]-module of rank n with determinant L. Suppose I is an ideal of A of height n such that there are two surjections α:P↠I and ϕ:L⊕An−1↠I. Assume that either (a) k=1 and n≥3 or (b) k is arbitrary but n≥4 is even. Then P has a unimodular element (see 4.1, 4.3). (2) Let R be a ring containing Q of even dimension n with height of the Jacobson radical of R≥2. Let P be a projective R[T,T−1]-module of rank n with trivial determinant. Assume that there exists a surjection α:P↠I, where I⊂R[T,T−1] is an ideal of height n such that I is generated by n elements. Then P has a unimodular element (see 3.4).

## First Page

2805

## Last Page

2814

## DOI

10.1016/j.jpaa.2017.01.009

## Publication Date

11-1-2017

## Recommended Citation

Keshari, Manoj K. and Ali Zinna, Md, "Existence of unimodular elements in a projective module" (2017). *Journal Articles*. 2373.

https://digitalcommons.isical.ac.in/journal-articles/2373

## Comments

Open Access, Green