Applications of Swan's Bertini to unimodular rows

Abstract

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

arXiv admin note: text overlap with arXiv:2104.09784