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.
[3] The norm residue isomorphism theorem was proved by Vladimir Voevodsky using a number of highly innovative results of Markus Rost.
denotes the Galois module of ℓ-th roots of unity in some separable closure of k. It induces an isomorphism
Taking the tensor products and applying the multiplicativity of étale cohomology yields an extension of the map
Specifically, Milnor K-theory is defined to be the graded parts of the ring: where
[4][5][6] 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.
[6][7] 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 case n = 2 and ℓ = 2 was proved by (Merkurjev 1981) harv error: no target: CITEREFMerkurjev1981 (help).
This case was proved by (Merkurjev & Suslin 1982) harv error: no target: CITEREFMerkurjevSuslin1982 (help) and is known as the Merkurjev–Suslin theorem.
Later, Merkurjev and Suslin, and independently, Rost, proved the case n = 3 and ℓ = 2 (Merkurjev & Suslin 1991) harv error: no target: CITEREFMerkurjevSuslin1991 (help) (Rost 1986) harv error: no target: CITEREFRost1986 (help).
, which takes values in the Brauer group of k (when the field contains all ℓ-th roots of unity).
These three properties implied, as a very special case, that the norm residue map should be an isomorphism.
In this case the strengthening that was needed required the development of a very large amount of new mathematics.
The earliest proof of Milnor's conjecture is contained in a 1995 preprint of Voevodsky[8] and is inspired by the idea that there should be algebraic analogs of Morava K-theory (these algebraic Morava K-theories were later constructed by Simone Borghesi[14]).
The constructions of 1995 and 1996 preprints are now known to be correct but the first completed proof of Milnor's conjecture used a somewhat different scheme.
Implementing this scheme required making substantial advances in the field of motivic homotopy theory as well as finding a way to build algebraic varieties with a specified list of properties.
From the motivic homotopy theory the proof required the following: The first two constructions were developed by Voevodsky by 2003.
Combined with the results that had been known since late 1980s, they were sufficient to reprove the Milnor conjecture.
Also in 2003, Voevodsky published on the web a preprint that nearly contained a proof of the general theorem.
The proof that they have the required properties was completed by Andrei Suslin and Seva Joukhovitski in 2006.
The third fact above required the development of new techniques in motivic homotopy theory.
The goal was to prove that a functor, which was not assumed to commute with limits or colimits, preserved weak equivalences between objects of a certain form.
One of the main difficulties there was that the standard approach to the study of weak equivalences is based on Bousfield–Quillen factorization systems and model category structures, and these were inadequate.
[citation needed] In the course of developing these techniques, it became clear that the first statement used without proof in Voevodsky's 2003 preprint is false.
The proof had to be modified slightly to accommodate the corrected form of that statement.
While Voevodsky continued to work out the final details of the proofs of the main theorems about motivic Eilenberg–MacLane spaces, Charles Weibel invented an approach to correct the place in the proof that had to modified.
Weibel also published in 2009 a paper that contained a summary of Voevodsky's constructions combined with the correction that he discovered.
This conjecture has now been proven, and is equivalent to the norm residue isomorphism theorem.