Nikita A. Karpenko and Alexander S. Merkurjev: On standard norm varieties

karpenko at, merkurev at

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 Hn+1(F,μp⊗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(YF(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 CH0(XL) 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.

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