On standard norm varieties

Details On standard norm varieties On standard norm varieties

2012-01-05

arxiv.org/abs/1201.1257v1

Mathematics

Algebraic Geometry

37 pages

Scientific paper

Let p be a prime integer and F a field of characteristic 0. Let X be the norm variety of a symbol in the Galois cohomology group H^{n+1}(F,\mu_p^{\otimes n}) (for some n>0), constructed in the proof of the Bloch-Kato conjecture. The main result of the paper affirms that the function field F(X) has the following property: for any equidimensional variety Y, the change of field homomorphism CH Y \to CH Y_{F(X)} of Chow groups with coefficients in integers localized at p is surjective in codimensions < (dim X)/(p-1). One of the main ingredients of the proof is a computation of Chow groups of a (generalized) Rost motive (a variant of the main result not relying on this is given in Appendix). Another important ingredient is A-triviality of X, the property saying that the degree homomorphism on CH_0 X_L is injective for any field extension L/F with non-empty X(L). The proof involves the theory of rational correspondences, due to Markus Rost, reviewed in Appendix.

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