Applications of Swan's Bertini to unimodular rows

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 $c.d.(k)\leq 1$.