Some remarks on orbit sets of unimodular rows

Abstract

Let $A$ be a $d$-dimensional smooth algebra over a perfect field of characteristic not $2$. Let $\mathrm{U}{m}_{n+1}(A)/{\mathrm{E}}_{n+1}(A)$ be the set of unimodular rows of length $n+1$ up to elementary transformations. If $n\ge (d+2)/2$, it carries a natural structure of group as discovered by van der Kallen. If $n=d\ge 3$, we show that this group is isomorphic to a cohomology group $H^d(A,G^{d+1})$. This extends a theorem of Morel, who showed that the set $\mathrm{U}{m}_{d+1}(A)/{\mathrm{SL}}_{d+1}(A)$ is in bijection with $H^d(A,G^{d+1})/{\mathrm{SL}}_{d+1}(A)$. We also extend this theorem to the case $d=2$. Using this, we compute the groups $\mathrm{U}{m}_{d+1}(A)/{\mathrm{E}}_{d+1}(A)$ when $A$ is a real algebra with trivial canonical bundle and such that $\mathrm{Spec}(A)$ is rational. We then compute the groups $\mathrm{U}{m}_{d+1}(A)/{\mathrm{SL}}_{d+1}(A)$ when $d$ is even, thus obtaining a complete description of stably free modules of rank $d$ on these algebras. We also deduce from our computations that there are no stably free non free modules of top rank over the algebraic real spheres of dimension $3$ and $7$.