We give an axiomatic framework for proving that the norm residue map
K^{M}_{n}(k)/l →
H^{n}(k,μ_{l}^{n})
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 {a_{1},...,a_{n}} of the
Milnor group K^{M}_{n}(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,e^{t}), and

2. The structure map y: M→χ to the Cech
variety χ and its dual Dy fit into complimentary triangles:

- 0809.bib (236 bytes)
- BK.dvi (43064 bytes) [December 18, 2006]
- BK.dvi.gz (17346 bytes)
- BK.pdf (136667 bytes)
- BK.ps.gz (168979 bytes)

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