Norm residue isomorphism theorem
In the mathematical field of algebraic K-theory, the norm residue isomorphism theorem is a long-sought result whose proof was completed in 2011. Its statement was previously known as the Bloch–Kato conjecture, after Spencer Bloch and Kazuya Kato, or more precisely the motivic Bloch–Kato conjecture in some places, since there is another Bloch–Kato conjecture on values of L-functions. The title "norm residue" originally referred to the symbol taking values in the Brauer group of k (when the field contains all ℓ-th roots of unity). Its usage here is in analogy with standard local class field theory which identifies the result in terms of a "higher" class field theory, still being developed.
It is a generalisation of the Milnor conjecture of K-theory, which was proved in the 1990s by Vladimir Voevodsky, the Milnor conjecture being the 2-primary part of the Bloch–Kato conjecture. The point of the conjecture is to equate the torsion in a K-group of a field F, algebraic information that is in general relatively inaccessible, with the torsion in a Galois cohomology group for F, which is in many cases much easier to compute. The Merkurjev–Suslin theorem is an intermediate result. Now that the complete proof of the Bloch–Kato conjecture has been announced, due to several mathematicians and contained in quite a number of papers, the result is also known as the Voevodsky–Rost theorem, for Voevodsky and Markus Rost.
For any integer ℓ invertible in a field k there is a map where denotes the group of ℓ-th roots of unity in some separable extension of k. It induces an isomorphism . By multiplicativity, the map ; extends to maps
with the property that vanishes. This is the defining relation of Milnor's definition of "higher" K-groups as the graded parts of the ring
The norm residue isomorphism theorem (or Bloch-Kato conjecture) states that for a field k and an integer ℓ that is invertible in k, the norm residue map
Let X be a smooth variety over a field containing . Beilinson and Lichtenbaum conjectured that the motivic cohomology group is isomorphic to the étale cohomology group when p≤q. This conjecture has now been proven, and is equivalent to the norm residue isomorphism theorem.
- Srinivas (1996) p.146
- Gille & Szamuely (2006) p.108
- Efrat (2006) p.221
- Srinivas (1996) pp.145-193
- Efrat, Ido (2006). Valuations, orderings, and Milnor K-theory. Mathematical Surveys and Monographs 124. Providence, RI: American Mathematical Society. ISBN 0-8218-4041-X. Zbl 1103.12002.
- Gille, Philippe; Szamuely, Tamás (2006). Central simple algebras and Galois cohomology. Cambridge Studies in Advanced Mathematics 101. Cambridge: Cambridge University Press. ISBN 0-521-86103-9. Zbl 1137.12001.
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- Voevodsky, Vladimir (2008). "On motivic cohomology with Z/l coefficients". arXiv:0805.4430 [math.AG].
- Weibel, Charles (2009). "The norm residue isomorphism theorem". Journal of Topology 2 (2): 346–372. doi:10.1112/jtopol/jtp013. MR 2529300.