On a question of Nori: obstructions, improvements, and applications

TL;DR

This paper investigates Nori's question on homotopy of projective modules over polynomial algebras, providing new results for smooth and singular cases, and introduces the Euler class group to study unimodular rows.

Contribution

It proves affirmative results for dimension two over algebraically closed fields, identifies obstructions in higher dimensions, and defines the Euler class group for affine algebras.

Findings

Affirmative answer for $ ext{dim}(R)=2$ over $ar{ ext{F}}_p$-algebras.

Identifies precise obstructions in the singular case for $ ext{dim}(R) ext{ } ext{geq} ext{ } 3$.

Defines the $n$-th Euler class group $E^n(A)$ and relates it to unimodular rows and stably free modules.

Abstract

This article concerns a question asked by M. V. Nori on homotopy of sections of Projective modules defined on the polynomial algebra over a smooth affine domain $R$. While this question has an affirmative answer, it is known that the assertion does not hold if: (1) $\dim(R)=2$; or (2) $d\geq 3$ but $R$ is not smooth. We first prove that an affirmative answer can be given for $\dim(R)=2$ when $R$ is an $\bar{\mathbb{F}}_p$-algebra. Next, for $d\geq 3$ we find the precise obstruction for the failure in the singular case. Further, we improve a result of Mandal (related to Nori's question) in the case when the ring $A$ is an affine $\bar{\mathbb{F}}_p$-algebra of dimension $d$. We apply this improvement to define the $n$-th Euler class group $E^n(A)$, where $2n\ge d+2.$ Moreover, if $A$ is smooth, we associate to a unimodular row $v$ of length $n+1$ its Euler class $e(v)\in E^n(A)$ and show that the corresponding stably free module, say, $P(v)$ has a unimodular element if and only if $e(v)$ vanishes in $E^n(A)$.