Let $k$ be a field (of char. not $2$) and $X\_k=\text{Spec} (k[x\_1,\cdots,x\_n]/(x\_1^2+\cdots +x\_n^2-1))$. Do we know the Chow groups $A\_i (X\_k)$? I could not find any references, even for $X\_{\mathbb R}$. 


What (I think) I know: the K-groups were computed by [Swan](http://www.jstor.org/pss/1971371), so we know the total Chow group up to torsions. In codimension $1$ (i.e., class groups) I am fairly certain the answers are known. 



Do we know the Chow groups of spheres?
The book


[The Algebraic and Geometric Theory of Quadratic Forms.](http://www.math.jussieu.fr/~karpenko/publ/Kniga.pdf) by R. Elman, N. Karpenko and A. Merkurjev (American Mathematical Society Colloquium Publications, 56., American Mathematical Society, Providence, RI, 2008. 435 pp.) 


contains a lot of information. I was just reading


B. Totaro, The automorphism group of an affine quadric, Math. Proc. Cambridge Philos. Soc. (2007) vol. 143 (1) pp. 1-8


and he is referring to this book for information on Chow groups of spheres.


