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.

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