Some remarks on orbit sets of unimodular rows

  • Jean Fasel

    Ecole Polytechnique Fédérale de Lausanne, Switzerland

Abstract

Let be a -dimensional smooth algebra over a perfect field of characteristic not . Let be the set of unimodular rows of length up to elementary transformations. If , it carries a natural structure of group as discovered by van der Kallen. If , we show that this group is isomorphic to a cohomology group . This extends a theorem of Morel, who showed that the set is in bijection with . We also extend this theorem to the case . Using this, we compute the groups when is a real algebra with trivial canonical bundle and such that is rational. We then compute the groups when is even, thus obtaining a complete description of stably free modules of rank 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 and .

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