Two preprints on Chow groups of quadrics, by Markus Rost

We make here available two older preprints on the K-cohomology of quadrics which play some role in the recent work of V. Voevodsky.

The preprint "On the spinornorm and A_0(X,K_1) for quadrics" ( from 1988 considers the group A_0(X,K_1) (which is the same thing as the K-cohomology group H^d(X,K_{d+1}), as well as the motivic cohomology group H^{2d+1}(X,Z(d+1)), with d=dimX) for quadrics X. It is shown that this group is always a quotient of the special Clifford group CL of the corresponding quadratic form. Meanwhile there is a precise result about the kernel of this map by A. S. Merkurjev and V. I. Chernousov: A_0(X,K_1) is the quotient of CL by the R-trivial elements in Spin.

Moreover we show that the norm map from A_0(X,K_1) to F^* is injective for certain quadrics including Pfister quadrics. This result is one of the inputs in Voevodsky's announced proof of the Bloch-Kato conjecture for p=2.

The preprint "Some new results on the Chowgroups of quadrics" (included here as chowqudr.dvi) from 1990 contains a brief description of the Chow motive corresponding to Pfister forms and their K-cohomology groups H^p(X,K_{p+i}) for i=0,1,2.

Some other preprints on the K-cohomology of quadrics can be found here.

Markus Rost <>