# Do we know the Chow groups of spheres?

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, 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.

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## 1 Answer

The book

The Algebraic and Geometric Theory of Quadratic Forms. 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.

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Thank you Andreas, this looks promising. I will try to look at it carefully when I have some time. –  Hailong Dao Oct 26 '10 at 2:11
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