### Some remarks on orbit sets of unimodular rows, by Jean Fasel

Let A be a smooth d-dimensional algebra over a perfect field of characteristic
different from 2. In this paper, we prove that the orbit set of unimodular rows
of length d+1 under elementary operations has a cohomological interpretation if
d is bigger or equal to 3. This allows to compute this set when A is a nice
smooth algebra over the field of real numbers. We then obtain a complete
description of the stably free modules of rank d over such an algebra, provided
that d is even.

Jean Fasel <jean.fasel@gmail.com>