Existence of unimodular elements in a projective module

TL;DR

This paper proves that under certain conditions on an affine algebra over an algebraically closed field of characteristic zero, projective modules of specific ranks over polynomial extensions contain unimodular elements.

Contribution

It establishes the existence of unimodular elements in projective modules over polynomial extensions of affine algebras under new rank and height conditions.

Findings

Unimodular elements exist in projective modules of rank n over certain polynomial extensions.

The result applies when the base algebra has dimension n and specific conditions on the number of variables and rank.

The theorem extends previous knowledge on projective modules and their unimodular elements in algebraic geometry.

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.