### Norm Varieties and the Chain Lemma (after Markus Rost), by Christian Haesemeyer and Charles A. Weibel

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,μ_{p}^{n})
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 p^{n-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.

- The Chain Lemma and the Norm Principle hold, by this paper.

- Given (1),
*Rost varieties* exist; this is proven in [SJ].
- Given (2),
*Rost motives* exist; this is proven in [V07] and [W07].
- Given (3), Bloch-Kato is true; this is proven in [V03] and [W06].

[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.

Christian Haesemeyer <chh@math.uic.edu>

Charles A. Weibel <weibel@math.rutgers.edu>