Norm residue isomorphism theorem

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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 (a_1,a_2) 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.

The norm residue isomorphism theorem implies the Quillen–Lichtenbaum conjecture. It is equivalent to a theorem whose statement was once referred to as the Beilinson–Lichtenbaum conjecture.

Statement[edit]

For any integer ℓ invertible in a field k there is a map \partial : k^*\rightarrow H^1(k,\mu_\ell) where \mu_\ell denotes the group of ℓ-th roots of unity in some separable extension of k. It induces an isomorphism k^*/k^{*\ell} \cong H^1(k,\mu_\ell). By multiplicativity, the map \partial; extends to maps

\partial^n : k^* \times \cdots \times k^* \rightarrow H^n\left({k,\mu_\ell^{\otimes n}}\right) \

with the property that \partial^n(\ldots,a,\ldots,1-a,\ldots) vanishes. This is the defining relation of Milnor's definition of "higher" K-groups as the graded parts of the ring

 K^M_*(k) := T^*(k^\times)/(a\otimes (1-a)) ,

the tensor algebra of the multiplicative group k× by the two-sided ideal, generated by the

\left \{a\otimes(1-a): \ a \neq 0,1 \right \}.

Hence \partial^n may be regarded as a map on K^M_n(k), called the Galois symbol or norm residue map.[1][2][3]

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

\partial^n : K_n^M(k)/\ell \to H^n_{{\acute{\rm e}{\rm t}}}(k,\mu^{\otimes n}_\ell)

from Milnor K-theory mod ℓ to étale cohomology is an isomorphism. The case ℓ=2 is the Milnor conjecture that was proved earlier by Voevodsky. The case n=2 is the Merkurjev–Suslin theorem.[3][4]

Beilinson-Lichtenbaum conjecture[edit]

Let X be a smooth variety over a field containing 1/\ell. Beilinson and Lichtenbaum conjectured that the motivic cohomology group H^{p,q}(X,Z/\ell) is isomorphic to the étale cohomology group H^p_{{\acute{\rm e}{\rm t}}}(k,\mu^{\otimes q}_\ell) when pq. This conjecture has now been proven, and is equivalent to the norm residue isomorphism theorem.

References[edit]

  1. ^ Srinivas (1996) p.146
  2. ^ Gille & Szamuely (2006) p.108
  3. ^ a b Efrat (2006) p.221
  4. ^ Srinivas (1996) pp.145-193