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 KnM(k)/p and
etale cohomology Hn(k,μpn)
for every prime p, every n and every field k containing 1/p.
Our proofs of these two results are based on 1998 Rost's preprints,
his web site and Rost's lectures at the Institute for Advanced Study
in 1999-2000 and 2005. By a Norm variety for a symbol, we will
mean a smooth projective p-generic splitting variety for the symbol
of dimension pn-1-1. Norm varieties exist by [SJ].
The proof of the Bloch-Kato conjecture has the following form.
Note that steps (1) and (2) are due to Rost, and that steps (3) and (4)
are due to Voevodsky.
[SJ] Suslin and Joukhovitski, Norm Varieties, JPAA 206 (2006).
[V03] Voevodsky, On motivic cohomology with Z/l-coefficients, 2003.
[V07] Voevodsky, Motivic Eilenberg-MacLane spaces, 2007.
[W06] Weibel, Axioms for the Norm Residue Isomorphism, 2006.
[W07] Weibel, Patching the Norm Residue Isomorphism Theorem, 2007.