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 K_2 of fields is given by the following theorem.

Matsumoto’s Theorem.
For any field F

K_{2}(F)= \frac{F^{*}\otimes_{\mathbb{Z}} F^{*}}{ <a \otimes (1-a)~|~ a\neq 1>}

or equivalently in the context of presentations of groups, K_{2}F is the Abelian group with
Generators: \{x,y\}\quad  x,y  \in F^{*}
Relations:

  1. \{x, 1-x\} =0\;     \forall x \in F^{*}   x \neq 0 (The Steinberg relation)
  2. \{xy,z\} = \{x,z\} +\{y,z\}\; \forall x,y, z \in F^{*}
  3. \{x, yz\} =\{x,y\} +\{x,z\}\; \forall x,y, z \in F^{*}.

Definition. Let F be a field and A be an Abelian group. A Steinberg symbol on F (with coefficients in A) is a \mathbb{Z}-bilinear map s:F^{*}\times F^{*}\longrightarrow A such that

s\{x,1-x\}=0\quad \forall x\in F^{*} \quad  x \neq 1.

By Matsumoto’s theorem any Steinberg symbol s:F^{*} \times F^{*} \longrightarrow A gives a unique group homomorphism \tilde{s}:K_2(F) \longrightarrow A such that s(x,y)=\tilde{s}\{x,y\}.

Norm Residue Algebras.

Let F be a field which contains a primitive n-th root of unity \omega and let \alpha,\beta be two given elements in F^{*}. The n^{2} dimensional F vector space

A_{\omega}(\alpha,\beta):= \bigoplus\limits_{0\leq i,j <n} F x^{i}y^{j}

with the following rules of multiplication:

x^{n}=\alpha \quad y^{n}=\beta \quad yx=\omega yx

is a central simple F-algebra and it is called the norm residue algebra.

Theorem 1. Let A be a central simple algebra of degree n and let

f(x)=x^{n}+a_{n-1}x^{n-1}+\dots +a_{0}

be the minimal polynomial of x\in A over F. If f splits into distinct linear factors over F, then A\simeq M_n(F).

Corollary. Let \alpha,\beta \in F^{*}. If either \alpha or \beta has an n-th root in F, then

A_{\omega}(\alpha,\beta)\simeq M_n(F).

As a special case of the above statement we have

A_{\omega}(\alpha,1)\simeq M_n(F).

Theorem 2. A_{\omega}(\alpha,1-\alpha)\simeq M_n(F)

Proof.
We define the non-commutative binomial coefficients

b_{i}^{n}:=\frac{f_n(c)}      {f_i(c)f_{n-i}(c)},

where

f_n(c)=\prod\limits_{j=1}^{n}(c^{j}-1)    .

It can be easily checked that b_{i}^{n}(c)\in \mathbb{Z}[c]. Now suppose that x,y are elements of an arbitrary ring R such that yx=cxy for some c in the center of R. Induction on n shows that

(x+y)^{n}= \sum_{i=0}^{n}b_{i}^{n}(c)x^{i}y^{n-i}.

In particular, for the generators x and y of A_{\omega}(\alpha,1-\alpha), since b_{0}^{n}(\omega)=b_{n}^{n}(\omega)=1 and b_{i}^{n}(\omega)=0 for all 0<i<n, we obtain that

(x+y)^{n}=x^{n}+y^{n}=\alpha+(1-\alpha)=1.

Now by the same reason as the previous corollary we have

A_{\omega}(\alpha,1-\alpha)\simeq M_n(F).

Theorem 3. Let \alpha, \beta, \gamma be in F^{*}. Then

A_{\omega}(\alpha,\beta)\otimes_F A_{\omega}(\alpha,\gamma)\simeq A_{\omega}(\alpha,\beta \gamma)\otimes_F A_{\omega}(1,\gamma).

Proof.
Let x_1, y_1 be the generators for A_{\omega}(\alpha,\beta) and x_2, y_2 be the generators for A_{\omega}(\alpha,\gamma). Define

x_3=x_1\otimes 1 \quad y_3=y_1\otimes y_2 \quad x_4=x_{1}^{-1}\otimes x_2 \quad y_4=1\otimes y_2.

Let A' be the algebra generated by x_3, y_3 and A'' be the algebra generated by x_4, y_4. Now x_{3}^{n}=\alpha\otimes 1, y_{3}^{n}=\beta\gamma\otimes 1 and

y_3x_3=y_1x_1\otimes y_2=\omega (x_1y_1\otimes y_2)=\omega x_3y_3.

So x_3 and y_3 satisfy the relations for A_{\omega}(\alpha,\beta \gamma), thus A'\simeq A_{\omega}(\alpha,\beta \gamma). Similarly A''\simeq A_{\omega}(1,\gamma). Notice that x_3 and y_3 commute with x_4 and y_4, hence we have a natural F-algebra homomorphism

\varphi:      A'\otimes_F A''\longrightarrow A_{\omega}(\alpha,\beta)\otimes_F A_{\omega}(\alpha,\gamma).

Since A'\otimes_F A'' is simple, \varphi is injective. Since the dimensions of two sides are equal n^{4} it must be an isomorphism.

Remark.
We have already seen that A_{\omega}(\alpha,1)\simeq M_n(F). So by the above theorem we have

[A_{\omega}(\alpha,\beta \gamma)]=[ A_{\omega}(\alpha,\beta)] [A_{\omega}(\alpha,\gamma)], similarly

[A_{\omega}(\alpha\beta, \gamma)]=[ A_{\omega}(\alpha,\gamma)][A_{\omega}(\beta,\gamma].

Here [A] denotes the equivalence class of A in the Brauer Group.

Define

s : F^{*}\times F^{*} \longrightarrow Br(F) \quad s(\alpha, \beta):= [A_{\omega}(\alpha,\beta)].

The above remark says that s is \mathbb{Z}-bilinear. By Theorem 3 we observe that s is a Steinberg symbol, hence we get a homomorphism

\tilde{s}: K_2(F)\longrightarrow Br (F) \quad \tilde{s}\{\alpha,\beta\}= [A_{\omega}(\alpha,\beta)].

From Corollary it follows that

[A_{\omega}(\alpha,\beta)]^{n}=[A_{\omega}(\alpha^{n},\beta)]=1,

which shows that the image of \tilde{s} is contained in

{_n}Br(F):=\{[A]\in Br(F)~|~[A]^{n}=1\}.

Since n\{\alpha,\beta\}=\{\alpha^{n},\beta\} , the homomorphism \tilde{s} annihilates nK_2(F), therefore it induces a homomorphism

R_{n,F}: K_2(F)/n K_2(F) \longrightarrow_n Br(F)

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 F be a field which contains an n-th primitive root of unity. Then

R_{n,F}: K_2(F)/nK_2(F)\longrightarrow  {_n}Br(F)

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 F be a field and let n be an integer coprime to char (F). Set

\mu_{n}=\{ x\in F_{sp} |\quad x^{n}=1 \}   .

The condition (n,char F)=1   implies that \mu_{n} has exactly n elements. Assume that F has an n-th primitive root of unity, i.e. \mu_{n}   \subset F. Set G:= Gal(F_{sp}/F) and consider the following exact sequence of G-modules:

1 \stackrel{\mu_{n}}{\rightarrow} \rightarrow F_{sp}^{*} \stackrel{\text{n}}{\rightarrow} F_{sp}^{*} \rightarrow1

The associated exact cohomology sequence is

1  \rightarrow H^{0} (G ,\mu_{n}) \rightarrow  H^{0} ( G ,F_{sp}^{*}) \stackrel{\text{n}}{\rightarrow}  H^{0}(G, F_{sp}^{*}) \rightarrow

H^{1} (G , \mu_{n}) \rightarrow  H^{1} (G ,F_{sp}^{*}) \stackrel{\text{ n}}{\rightarrow} H^{1}(G, F_{sp}^{*}) \rightarrow

H^{2} ( G ,\mu_{n}) \rightarrow  H^{2} ( G ,F_{sp}^{*}) \stackrel{\text{n}}{\rightarrow}  H^{2}(G , F_{sp}^{*}).

As \mu_{n} \subset F, the action of G on \mu_{n} is trivial, so H^{0} (G ,\mu_{n})=\mu_{n} . By Hilbert’s Satz 90 we have H^{1} ( G ,F_{sp}^{*})=1, so the above sequence breaks up to the following exact sequences:

1 \rightarrow \mu_{n} \rightarrow F^{*} \stackrel{\text{n}} {\rightarrow} F^{*} \stackrel{\delta}{\rightarrow} H^{1}(G,\mu_{n}) \rightarrow 1

1 \rightarrow H^{2}(G , \mu_{n}) \stackrel{\lambda} {\rightarrow} H^{2}(G , F_{sp}) \stackrel{\text{n}}{\rightarrow}  H^{2}(G , F_{sp}).

Hence the map \delta induces an isomorphism H^{1}(G , \mu_{n})\simeq F^{*}/F^{*^{n}}) , and the map \lambda induces an isomorphism between

H^{2}(G , \mu_{n}) and ker( H^{2}(G , F_{sp}) \stackrel{\text{ n}}{\rightarrow}  H^{2}(G ,F_{sp})).

By using that Br(F) \simeq      H^{2}(G , F_{sp}) we obtain that H^{2}(G , \mu_{n})\simeq _nBr(F). Since G acts trivially on \mu_{n} it follows that \mu_{n}^{\otimes^{2}} is isomorphic to \mu_{n} as G-module, hence

H^{2}(G ,    \mu^{\otimes^{2}} ) \simeq             H^{2}(G , \mu_{n}) \simeq{_n}Br(F).

The composition of the following maps

F^{*} \times F^{*} \rightarrow F^{*}/F^{*^{n}}\times F^{*}/F^{*^{n}}\simeq H^{1}(G,\mu_{n})\times H^{1}(G,\mu_{n}) \stackrel{\cup}{\rightarrow} H^{2}(G, \mu_{n})\simeq  {_n}Br(F)

gives a \mathbb{Z}-bilinear map which can be proved to be a Steinberg symbol, and the induced homomorphism K_2(F)\longrightarrow {_n}Br(F) is the norm residue homomorphism.

References:

1. Kersten, Ina Brauergruppen von K├Ârpern. (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.

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