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 ≥ (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 \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 .