Applications of Swan's Bertini to unimodular rows

TL;DR

This paper explores the algebraic structure of unimodular rows over affine algebras using Swan's Bertini theorem, revealing conditions under which certain quotient groups are well-behaved.

Contribution

It applies Swan's Bertini to establish new structural properties of unimodular row groups over affine algebras in various field conditions.

Findings

Group structures are well-behaved under specific field and algebra conditions.

Unimodular row groups are uniquely divisible in certain cases.

Results depend on the cohomological dimension of the base field.

Abstract

Let $R$ be an affine algebra of dimension $d\geq 4$ over a perfect field $k$ of char $\neq 2$ and $I$ be an ideal of $R$. Then - Um$_{d+1}(R,I)/{\rm E}_{d+1}(R,I)$ has nice group structure if $c.d._2(k)\leq 2$. - Um$_d(R,I)/{\rm E}_d(R,I)$ has nice group structure if $k$ is algebraically closed of char $k\neq 2,3$ and either (i) $k = \overline{\mathbb{F}}_{p}$ or (ii) $R$ is normal. - $MS_{d+1}(R)$ is uniquely divisible prime to characteristic of $k$ if $R$ is reduced and $k$ is infinite with $c.d.(k)\leq 1$.