On standard norm varieties

TL;DR

This paper proves a key property of norm varieties related to Chow groups and their surjectivity over field extensions, advancing understanding of algebraic cycles in the context of the Bloch-Kato conjecture.

Contribution

It establishes a surjectivity property of Chow groups over the function field of norm varieties, utilizing computations of Chow groups of Rost motives and the concept of $A$-triviality.

Findings

Chow groups over the function field of norm varieties are surjective in certain codimensions.

Computed Chow groups of generalized Rost motives.

Established $A$-triviality of norm varieties.

Abstract

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 $H^{n+1}(F,\mu_p^{\otimes n})$ (for some $n\geq1$), 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 {\em $A$-triviality} of $X$, the property saying that the degree homomorphism on $\CH_0(X_L)$ is injective for any field extension $L/F$ with $X(L)\ne\emptyset$. The proof involves the theory of rational correspondences reviewed in Appendix.