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

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 $E^n(A)$.