karpenko at math.jussieu.fr, merkurev at math.ucla.edu

Submission: 2012, Jan 4

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,μ_{p}^{⊗n}) (for some n≥1), 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)→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
X(L)≠∅.
The proof involves
the theory of rational correspondences, due to Markus Rost, reviewed in Appendix.

2010 Mathematics Subject Classification: 14C25

Keywords and Phrases: Norm varieties, Chow groups and motives, Steenrod operations.

Full text: dvi.gz 85 k, dvi 222 k, ps.gz 1044 k, pdf.gz 358 k, pdf 438 k.

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