For $n \in \mathbb{ N}$ and a commutative ring $R$ with $2 \in R^{ \times}$, the group $SL_n (R)$ acts on the set $Um_n (R)$ of unimodular vectors of length $n$ and $Spin_{ 2n}(R)$ acts on the set of unit vectors $U_{ 2n-1}(R)$. We give an example of a ring for which the comparison map $Um_n (R)/SL_n (R) \rightarrow U_{ 2n-1}(R)/Spin_{ 2n}(R)$ fails to be bijective.
Categories: math.AG math.AC math.KT