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 $F$

$K_{2}(F)= \frac{F^{*}\otimes_{\mathbb{Z}} F^{*}}{ }$

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

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, 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 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.