Recently, the norm-residue homomorphism has been the subject of intense discussions in the K-theoretic community following the proof of the Bloch-Kato conjecture by Voevodsky, Suslin and Rost (see Rost’s lecture at this year’s Arbeitstagung.) The goal of this post is to explain the norm-residue homomorphism in a down to earth ring-theoretic language.
Recall that the description of of fields is given by the following theorem.
For any field
or equivalently in the context of presentations of groups, is the Abelian group with
- (The Steinberg relation)
Definition. Let be a field and be an Abelian group. A Steinberg symbol on (with coefficients in ) is a -bilinear map such that
By Matsumoto’s theorem any Steinberg symbol gives a unique group homomorphism such that
Norm Residue Algebras.
Let be a field which contains a primitive -th root of unity and let be two given elements in . The dimensional vector space
with the following rules of multiplication:
is a central simple -algebra and it is called the norm residue algebra.
Theorem 1. Let be a central simple algebra of degree and let
be the minimal polynomial of over . If splits into distinct linear factors over , then
Corollary. Let . If either or has an -th root in , then
As a special case of the above statement we have
We define the non-commutative binomial coefficients
It can be easily checked that . Now suppose that are elements of an arbitrary ring such that for some in the center of . Induction on shows that
In particular, for the generators and of , since and for all , we obtain that
Now by the same reason as the previous corollary we have
Theorem 3. Let be in . Then
Let be the generators for and be the generators for . Define
Let be the algebra generated by and be the algebra generated by . Now , and
So and satisfy the relations for , thus . Similarly . Notice that and commute with and , hence we have a natural -algebra homomorphism
Since is simple, is injective. Since the dimensions of two sides are equal it must be an isomorphism.
Remark. We have already seen that . So by the above theorem we have
Here denotes the equivalence class of in the Brauer Group.
The above remark says that is -bilinear. By Theorem 3 we observe that is a Steinberg symbol, hence we get a homomorphism
From Corollary it follows that
which shows that the image of is contained in
Since , the homomorphism annihilates , therefore it induces a homomorphism
which is called the norm residue homomorphism.
The following surprising theorem was proved by A. Merkurjev and
A. Suslin in 1982.
The Merkurjev-Suslin Theorem. Let be a field which contains an -th primitive root of unity. Then
is an isomorphism.
Norm Residue Homomorphism via Galois Cohomology.
The norm residue homomorphism can be described in terms of Galois cohomology. As a preliminary we need to recall the notion of the cup product in the cohomology of groups.
Let be a field and let be an integer coprime to char Set
The condition implies that has exactly elements. Assume that has an -th primitive root of unity, i.e. . Set and consider the following exact sequence of -modules:
The associated exact cohomology sequence is
As , the action of on is trivial, so . By Hilbert’s Satz we have , so the above sequence breaks up to the following exact sequences:
Hence the map induces an isomorphism , and the map induces an isomorphism between
By using that we obtain that Since acts trivially on it follows that is isomorphic to as -module, hence
The composition of the following maps
gives a -bilinear map which can be proved to be a Steinberg symbol, and the induced homomorphism is the norm residue homomorphism.
2. Milnor, John Introduction to algebraic $K$-theory. Annals of Mathematics Studies, No. 72.
3. Rost, Markus Arbeitstagung 2007 – Norm residue homomorphism.