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Concept
Norm residue isomorphism theorem
Summary
In mathematics, the norm residue isomorphism theorem is a long-sought result relating Milnor K-theory and Galois cohomology. The result has a relatively elementary formulation and at the same time represents the key juncture in the proofs of many seemingly unrelated theorems from abstract algebra, theory of quadratic forms, algebraic K-theory and the theory of motives. The theorem asserts that a certain statement holds true for any prime and any natural number . John Milnor speculated that this theorem might be true for and all , and this question became known as Milnor's conjecture. The general case was conjectured by Spencer Bloch and Kazuya Kato and became known as the Bloch–Kato conjecture or the motivic Bloch–Kato conjecture to distinguish it from the Bloch–Kato conjecture on values of L-functions. The norm residue isomorphism theorem was proved by Vladimir Voevodsky using a number of highly innovative results of Markus Rost.
For any integer l invertible in a field there is a map
where denotes the Galois module of l-th roots of unity in some separable closure of k. It induces an isomorphism . The first hint that this is related to K-theory is that is the group K1(k). Taking the tensor products and applying the multiplicativity of étale cohomology yields an extension of the map to maps:
These maps have the property that, for every element a in , vanishes. This is the defining relation of Milnor K-theory. Specifically, Milnor K-theory is defined to be the graded parts of the ring:
where is the tensor algebra of the multiplicative group and the quotient is by the two-sided ideal generated by all elements of the form . Therefore the map factors through a map:
This map is called the Galois symbol or norm residue map. Because étale cohomology with mod-l coefficients is an l-torsion group, this map additionally factors through .
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
from Milnor K-theory mod-l to étale cohomology is an isomorphism.
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