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.

**Matsumoto’s Theorem.**

For any field

or equivalently in the context of presentations of groups, is the Abelian group with

**Generators:**

**Relations:**

- (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

.

**Theorem 2**.

**Proof.**

We define the non-commutative binomial coefficients

where

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

**Proof.**

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

similarly

Here denotes the equivalence class of in the Brauer Group.

Define

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

and

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.

**References: **

1. Kersten, Ina Brauergruppen von Körpern. (German) [Brauer groups of fields] Aspects of Mathematics, D6.1.

2. Milnor, John Introduction to algebraic $K$-theory. Annals of Mathematics Studies, No. 72.

3. Rost, Markus Arbeitstagung 2007 – Norm residue homomorphism.** **

4. Tate, John Relations between $K\sb{2}$ and Galois cohomology. *Invent. Math.* ** 36 ** (1976), 257–274.

## 9 comments

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July 12, 2007 at 5:50 pm

davidspeyerI think you have a typo. Three lines above the statement of Theorem 3, you want to say that (x+y)^n=x^n+y^n=alpha+(1-alpha)=1, not =0, right?

July 12, 2007 at 7:26 pm

AntonThank you, I’ve corrected that. And also instead of . 🙂

July 16, 2007 at 3:17 pm

Jason StarrThis reminds me of another theorem of Merkurjev-Suslin: if F is a perfect field of cohomological dimension 2, then for every central simple algebra A over F the reduced norm map Nm:A^* –> F^* is surjective. With no hypotheses on F, it is not hard to see the cokernel depends only on [A] in Br(F) and factors through F^*/(F^*)^n, where n = order([A]). Thus, given [A] in Br(F)[n]=H^2(F,\mu_n), the cokernel is a quotient of H^1(F,\mu_n). It is reasonable to guess the cokernel is the image of the homomorphism H^1(F,\mu_n) –> H^3(F,\mu_n^2) determined by cup product with [A]. If so, that would explain this MS theorem, since H^3(F,\mu_n^2) vanishes if cd(F)=2. Do you know if this is correct? If so, is the proof related to the MS theorem in your post?

July 17, 2007 at 1:55 pm

AntonHi,

probably you are right may be you can use the MS theorem above to prove that the image of the norm map in is exactly the part killed by the cup product with . For example, if then is the norm of and indeed, . But I don’t know much about the norm map, so we should look in the Merkur’jev-Suslin papers.

July 18, 2007 at 11:13 pm

Jason StarrHi,

I will take a look. I ran across a comment in Serre’s “Galois cohomology” that MS construct a map from the cokernel of the reduced norm map into H^3(F,\mu_n^2) <>. That hypothesis makes me believe there is more to this than meets the eye.

Best,

Jason

July 19, 2007 at 12:20 pm

Jason StarrThe hypothesis that was supposed to appear in “” somehow didn’t compile: MS construct the map if n is square-free.

May 27, 2008 at 6:07 am

BenPARI is already integrated with Python:

http://www.sagemath.org

May 27, 2008 at 6:10 am

BenSorry, my comment above was not meant to be on this post.

August 30, 2009 at 6:43 am

PEN ChentraDear Madams/Miss/Mrs

Can you help me to proof (Z/abZ) and (Z/aZ)(Z/bZ) is homomorphisme (gcd(a,b)=1).

Sincerely