# Commentarii Mathematici Helvetici

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**Volume 86, Issue 1, 2011, pp. 13–39**

**DOI: 10.4171/CMH/216**

Published online: 2010-12-05

Some remarks on orbit sets of unimodular rows

Jean Fasel^{[1]}(1) Ecole Polytechnique Fédérale de Lausanne, Switzerland

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`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.

*Keywords: *Unimodular rows, Witt and Grothendieck–Witt groups, Milnor–Witt K-theory

Fasel Jean: Some remarks on orbit sets of unimodular rows. *Comment. Math. Helv.* 86 (2011), 13-39. doi: 10.4171/CMH/216