2012/01/05 by Nikita A. Karpenko, Karpenko, Nikita A., Alexander S. Merkurjev +1
Mathematics · #14C25 #Algebraic Geometry (math.AG) #FOS: Mathematics #math.AG #msc:14C25
paper · pdf · doi:10.48550/arxiv.1201.1257
38 pages; final version, to appear in Ann. Sci. Éc. Norm. Supér. (4)
arxiv created 2012/06/19 · arxiv updated 2012/06/20
Let p be a prime integer and F a field of characteristic 0. Let X be the \em 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 \em 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 reviewed in Appendix.