论文标题
关于诺里的问题：障碍，改进和应用
On a question of Nori: obstructions, improvements, and applications
论文作者
论文摘要
本文涉及M. V. Nori对多项式代数上定义的投影模块的各个部分的同喻，该部分在平滑的仿射域$ r $上。虽然这个问题有一个肯定的答案，但众所周知，当$ \ dim（r）= 2 $;或（2）$ d \ geq 3 $，但$ r $并不光滑。我们首先证明，当$ r $是$ \ bar {\ mathbb {f}} _ p $ -Algebra时，可以为$ \ dim（r）= 2 $给出肯定答案。接下来，对于$ d \ geq 3 $，我们找到了单个情况下故障的确切阻塞。此外，在戒指$ a $是affine $ \ bar {\ mathbb {f}} _ p $ - dimension $ d $的情况下，我们改善了曼达尔（与Nori的问题有关）的结果（与Nori的问题有关）。 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只有$ e（v）$在$ e^n（a）$中消失。
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 $E^n(A)$.
