Do we know the Chow groups of spheres? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T00:22:00Z http://mathoverflow.net/feeds/question/43535 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/43535/do-we-know-the-chow-groups-of-spheres Do we know the Chow groups of spheres? Hailong Dao 2010-10-25T15:57:16Z 2010-10-25T19:22:36Z <p>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 $X_{\mathbb R}$. </p> <p>What (I think) I know: the K-groups were computed by <a href="http://www.jstor.org/pss/1971371" rel="nofollow">Swan</a>, 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. </p> http://mathoverflow.net/questions/43535/do-we-know-the-chow-groups-of-spheres/43562#43562 Answer by Andreas Thom for Do we know the Chow groups of spheres? Andreas Thom 2010-10-25T19:22:36Z 2010-10-25T19:22:36Z <p>The book</p> <p><a href="http://www.math.jussieu.fr/~karpenko/publ/Kniga.pdf" rel="nofollow">The Algebraic and Geometric Theory of Quadratic Forms.</a> by R. Elman, N. Karpenko and A. Merkurjev (American Mathematical Society Colloquium Publications, 56., American Mathematical Society, Providence, RI, 2008. 435 pp.) </p> <p>contains a lot of information. I was just reading</p> <p>B. Totaro, The automorphism group of an affine quadric, Math. Proc. Cambridge Philos. Soc. (2007) vol. 143 (1) pp. 1-8</p> <p>and he is referring to this book for information on Chow groups of spheres.</p>