Axioms for the Norm Residue Isomorphism, by Charles A. Weibel

We give an axiomatic framework for proving that the norm residue map KMn(k)/l → Hn(k,μln) is an isomorphism (i.e., for settling the motivic Bloch-Kato conjecture). This framework is a part of the Voevodsky-Rost program, and is an axiomatic reworking of the ending of Voevodsky's 2003 preprint, based upon a suggestion of Rost.

The main theorem states that, if we assume that the norm residue map is an isomorphism in degrees < n, and if for every nonzero element {a1,...,an} of the Milnor group KMn(k)/l, there is a direct summand M of a Rost variety X satisfying the axioms below, then the norm residue map is an isomorphism in degree n.

Here are the axioms that M must satisfy:
1. As a Chow motive, the direct summand M=(X,e) is isomorphic to (X,et), and
2. The structure map y: M→χ to the Cech variety χ and its dual Dy fit into complimentary triangles:

D ⊗ Lb → M → M(χ) →,
M(χ)⊗ Lb → M → D →.


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