Norm Varieties and the Chain Lemma (after Markus Rost)

TL;DR

This paper provides proofs of Markus Rost's Chain Lemma and Norm Principle, which are crucial steps in verifying the Bloch-Kato conjecture relating Milnor K-theory and etale cohomology.

Contribution

It offers the final proofs of two key results needed to confirm the Bloch-Kato conjecture, completing a major step in algebraic K-theory and Galois cohomology.

Findings

Proof of the Chain Lemma

Proof of the Norm Principle

Verification of the Bloch-Kato conjecture

Abstract

The goal of this paper is to present proofs of two results of Markus Rost: the Chain Lemma and the Norm Principle. These are the final steps needed to complete the publishable verification of the Bloch-Kato conjecture, that the norm residue maps are isomorphisms between Milnor K-theory $K_n^M(k)/p$ and etale cohomology $H^n(k,\mu_p^n)$ for every prime p, every n and every field k containing 1/p. Our proofs of these two results are based on Rost's 1998 preprints, his web site and Rost's lectures at the Institute for Advanced Study in 1999-2000 and 2005.