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.
Statement
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. The case is the Milnor conjecture, and the case is the Merkurjev–Suslin theorem.
History
The étale cohomology of a field is identical to Galois cohomology, so the conjecture equates the ℓth cotorsion (the quotient by the subgroup of ℓ-divisible elements) of the Milnor K-group of a field k with the Galois cohomology of k with coefficients in the Galois module of ℓth roots of unity. The point of the conjecture is that there are properties that are easily seen for Milnor K-groups but not for Galois cohomology, and vice versa; the norm residue isomorphism theorem makes it possible to apply techniques applicable to the object on one side of the isomorphism to the object on the other side of the isomorphism.
The case when n is 0 is trivial, and the case when follows easily from Hilbert's Theorem 90. The case and was proved by . An important advance was the case and ℓ arbitrary. This case was proved by and is known as the Merkurjev–Suslin theorem. Later, Merkurjev and Suslin, and independently, Rost, proved the case and .
The name "norm residue" originally referred to the Hilbert symbol , which takes 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 and is expected to be part of an (as yet undeveloped) "higher" class field theory.
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.
History of the proof
Milnor's conjecture was proved by Vladimir Voevodsky. Later Voevodsky proved the general Bloch–Kato conjecture.