On a question of Moshe Roitman and Euler class of stably free module

TL;DR

This paper investigates conditions under which projective modules over polynomial rings are stably free, extending previous results and addressing a question posed by Moshe Roitman and Euler class considerations.

Contribution

It establishes new criteria for the existence of unimodular elements in projective modules over polynomial extensions of rings, generalizing prior work and answering open questions.

Findings

Proves that under certain dimension and extension conditions, projective modules have unimodular elements.

Extends results to rings over algebraically closed fields of positive characteristic.

Provides conditions linking the Euler class of modules to their stably free status.

Abstract

Let $A$ be a ring of dimension $d$ containing an infinite field $k$, $T_1,\ldots,T_r$ be variables over $A$ and $P$ be a projective $A[T_1,\ldots,T_r]$-module of rank $n$. Assume one of the following conditions hold. (1) $2n\geq d+3$ and $P$ is extended from $A$. (2) $2n\geq d+2$, $A$ is an affine $\overline {\mathbb F}_p$-algebra and $P$ is extended from $A$. (3) $2n\geq d+3$ and singular locus of $Spec(A)$ is a closed set $V(\mathcal J)$ with ht $\mathcal J\geq d-n+2$. Assume $Um(P_f)\neq \varnothing$ for some monic polynomial $f(T_r)\in A[T_1,\ldots,T_r]$. Then $Um(P)\neq \varnothing$.