# Some remarks on orbit sets of unimodular rows

### Jean Fasel

Ecole Polytechnique Fédérale de Lausanne, Switzerland

## Abstract

Let $A$ be a $d$-dimensional smooth algebra over a perfect field of characteristic not $2$. Let $Um_{n+1}(A)/E_{n+1}(A)$ be the set of unimodular rows of length $n +1$ up to elementary transformations. If $n≥(d +2)/2$, it carries a natural structure of group as discovered by van der Kallen. If $n=d≥3$, 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 $Um_{d+1}(A)/SL_{d+1}(A)$ is in bijection with $H_{d}(A,G_{d+1})/SL_{d+1}(A)$. We also extend this theorem to the case $d=2$. Using this, we compute the groups $Um_{d+1}(A)/E_{d+1}(A)$ when $A$ is a real algebra with trivial canonical bundle and such that $Spec (A)$ is rational. We then compute the groups $Um_{d+1}(A)/SL_{d+1}(A)$ 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 $7$.

## Cite this article

Jean Fasel, Some remarks on orbit sets of unimodular rows. Comment. Math. Helv. 86 (2011), no. 1, pp. 13–39

DOI 10.4171/CMH/216