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\def \dfun {\frak G}
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\topmatter
\title
 $SK_1$-like functors for division algebras
\endtitle

%\rightheadtext{ }
\author
R. Hazrat
\endauthor

\address
Department of Mathematics,
University of Bielefeld,
P. O. Box 100131, 33501 Bielefeld,
Germany.
\endaddress

\email
rhazrat$\@$mathematik.uni-bielefeld.de
\endemail

\abstract
{We introduce a Dieudonne/reduced Dieudonne functor for central simple 
algebras $A$, which embraces different groups which are associated to $A$ in 
different contexts, including the reduced Whitehead group $SK_1(A)$.  
We show that a reduced Dieudonne functor has 
most important functorial properties of the reduced 
Whitehead group $SK_1(A)$. 
Specializing a Dieudonne functor to the group $G(A)=A^*/F^*A'$,
where $F$ is the center of $A$ and $A'$ its commutator subgroup,
we establish a fundamental connection between this group, its residue 
version and 
relative valued group when $A$ is a henselian division algebra. The structure
of $G(A)$ turns out to carry significant information about the arithmetic
of $A$. Along these lines, we employ $G(A)$ to compute the group $SK_1(A)$.
As an application, we obtain theorems of reduced $K$-theory which requires
heavy machinery, as simple examples of our method.

\bigskip

\noindent AMS Classification  12E15.}
\endabstract
\keywords 
 Division ring, Reduced $K$-theory
\endkeywords

\endtopmatter

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\document

\sec {Introduction}

\bigskip

Let $D$ be a division algebra with center $F$. 
The non-triviality of the important group 
$SK_1(D)$ is shown by V. P. Platonov who developed a so-called 
{\it Reduced K-Theory} to compute $SK_1(D)$ for certain division algebras. 
The group $SK_1(D)$ enjoys some interesting properties which makes it 
distinguish from the {\it K-Theory} functor $K_1(D)$. Among interesting
characteristic of the torsion group $SK_1(D)$ is its behavior under extension of 
the ground field. Also in the case of valued division algebra, 
its stability  
under reduction, namely
$SK_1(D) = SK_1( \overline D)$, where $D$ is unramified division algebra.
Also the primary decomposition of a division algebra induces a corresponding 
decomposition of $SK_1(D)$.
 

On the other hand, there have been other groups which are associated with a 
division algebra $D$, in order to study the group and arithmetic structure of 
$D$, e.g, $D^*/Nrd(D^*)D'$ in [1], or $D^*/{D^*}^2$ in [8]. Also see [5].
It can be seen that these groups are in close connection with the group $SK_1(D)$.

On this note we introduce a ``Dieudonne'' functor for division algebras which
embraces groups that have been already associated with division algebra in 
different contexts. We then show that a ``reduced'' Dieudonne functor 
(See Definition 2.2) shares most important functorial properties of the 
reduced Whitehead group.
Notice that a reduced Dieudonne functor covers a wide range of groups of 
different nature.
In section 2 we will 
show that a reduced Dieudonne functor is torsion of bounded exponent.
we then obtain some $SK_1$-like properties for this functor. 
we show that a reduced Dieudonne functor may grow 
``pathologically'' for algebraic extension of ground field whose degree is prime
to the index of $D$. It is then shown that  
this functor satisfies a decomposition property analog to one for $SK_1(D)$
(Theorem 2.10).


We then specialize a Dieudonne functor to the group $G(D)=D^*/F^*D'$
where $D$ is a division algebra over its center $F$ and $D'$ is its 
commutator subgroup.
It turns out that there is a close connection between the group structure of 
$G(D)$ and algebraic structure of $D$. For example in section 3, after 
establishing a fundemental connection between $G(D)$, its residue 
version and relative valued group when $D$ enjoys a henselian valuation,
we show that if $D$ is a totally ramified division
algebra, then there is a one to one correspondence between isomorphism classes
of $F$-subalgebra of $D$ and the subgroups of $G(D)$.
We then use $G(D)$ to compute $SK_1(D)$ for certain division algebras.
We show that if $G(D)$ canonically coincides with the relative valued group,
then there is a explicit formula for the group $SK_1(D)$ (Theorem 3.7). 
It turns out that 
many theorems and examples of reduced $K$-theory which require heavy
machinery can all be viewed as simple examples of our case. 
Section 3 is devoted to the unitary version of the group $G(D)$.

We fix some notations. Let $D$ be a division algebra over its center $F$ with 
index $i(D)=n$. Then $Nrd_{D/F}:D^*
 \longrightarrow F^*$
is the reduced norm function and $SK_1(D)=D^{(1)}/D'$ is the reduced Whitehead 
group where $D^{(1)}$ is the kernel of $Nrd_{D/F}$. Put $SH^0(D)$
for the cokernel of $Nrd_{D/F}$.
we take $\mu_n(F)$  for the group of $n-th$ roots of unity in $F$, and $Z(D')$ 
for the center of the group $D'$. Observe that $\mu_n(F)=F^* \cap D^{(1)}$ and 
$Z(D')=F^* \cap D'$. If $G$ is a group, by $G^n$ we denote the subgroup of $G$
generated by all elements of $n-th$ power of $G$. Let $exp(G)$ stands for the 
exponent of the group $G$.
Let also $det:GL_n(D)/SL_n(D) \longrightarrow D^*/D'$ denote the Dieudonne determinant, where $GL_n(D)$ is the general linear group and $SL_n(D)$ is its 
commutator subgroup. 


\bigskip

\sec {Dieudonne Functor}

\bigskip


Let $\Cal C$ be the class of all central simple algebras and 
$\dfun : \Cal C \longrightarrow \Cal Ab $ be a functor from $\Cal C$ to the
category of abelian groups.


\define The functor $\dfun$ is called a {\it Dieudonne functor} 
if for any division algebra $D$ and a nonnegative integer $n$, 
there is a homomorphism
$d :\dfun(M_n(D)) \longrightarrow \dfun (D)$ such that for any 
$x \in \dfun(A)$, 
$di(x)=x^n$, where $i:\dfun(D) \longrightarrow \dfun(M_n(D))$ induced 
by the natural embedding $D \longrightarrow M_n(D)$.  

The functor $K_1$ with the Dieudonne determinant $det$ as $d$,
clearly forms a Dieudonne functor.  

Let $A$ be a central simple algebra and $(A^*)^2$ be the 
subgroup of $A^*$ generated by the squares of $A^*$. It is easy to see that 
$\dfun(A)=A^*/(A^*)^2$ with $det$ as above is a Dieudonne functor. 
This group is studied in [8] in 
connection with Witt rings of division algebras.

\define  \label{dieudonne}
A Dieudonne functor $\dfun $ is said to be {\it reduced} if for any 
field $F$, $\dfun(F)$ is trivial and if $x \in Ker(\dfun(M_n(D)) 
\longrightarrow \dfun(D))$, where $D$ is a division algebra and 
$n \in \Bbb N$, then $x^n=1$. 

\example
Let $A \in \Cal C$ with center 
$F$, then the functors $\dfun(A)=A^*/F^*A'$, $\dfun(A)=(A^*)^2/(F^*)^2A'$, 
and $\dfun(A)=A^*/F^*A'_r$ where $A'_r= \{ x\in A^* | x^r \in A'\}$ 
and $r \in \Bbb N$  can be all viewed as reduced Dieudonne functors.

\example
Let $A \in \Cal C$ be a central simple algebra finite over its center.
In order to consider the reduced Whitehead group $SK_1(A)$ and $SH^0(A)$ 
as a reduced Dieudonne functors, we shall limit the morphisms of our 
category (See chapters 22 and 23 in [3]). The following commutative diagram 
shows that 
$SK_1$ and $SH^0$ are Dieudonne functors.
$$\diagram
1 \rto & SK_1(D) \dto \rto & K_1(D) \rto^{Nrd_{D/F}} \dto_{} & F^* \dto^{\eta_n} \rto & SH^0(D) \rto \dto &1 \\
 1 \rto & SK_1(M_n(D)) \dto \rto & K_1(M_n(D)) \rto^{ \qquad Nrd_{D/F}} \dto_{det} & F^* \dto^{1} \rto & SH^0(M_n(D)) \rto \dto &1 \\
1 \rto & SK_1(D) \rto & K_1(D) \rto^{Nrd_{D/F}} & F^*  \rto & SH^0(D) \rto  &1
\enddiagram$$
where $\eta_n(x)=x^n$ for any $x \in F^*$ and $D$ is a division algebra with
center $F$. Now it is easy to see that $SK_1(A)=A^{(1)}/A'$ and 
$SH^0(A)=F^*/Nrd_{A/F}(A^*)$ satisfy conditions of being reduced Dieudonne 
functor.

In the same way, it can be seen that $\dfun(A)=A^*/Nrd_{A/F}(A^*)A'$ and
$\dfun(A)=A^*/F^*A^{(1)} \simeq Nrd(A^*)/{F^*}^{Deg A}$ are also reduced Dieudonne functors.

Our primary aim in this section is to show that a reduced Dieudonne functor 
$\dfun$ shares almost all important functorial properties of $SK_1$.
Note that a reduced Dieudonne functor covers groups of different nature which
are associated to a division algebra. We will show that this functor is 
torsion of bounded exponent.  

We begin by showing that a reduced Dieudonne functor may grow pathologically 
for algebraic extension of ground field whose degree is prime to the index of
division algebra $D$.

Clearly the natural embedding of $D$ in $D \otimes_F L$ where $L$ is a finite 
field extension of $F$, induce a group 
homomorphism $\Cal I : \dfun(D) \longrightarrow \dfun(D \otimes_F L)$.
The following proposition provides us with a homomorphism in the opposite
direction.



\prop \label{DtensorL}
Let $\dfun$ be a Dieudonne functor and $D$ be a division ring with center 
$F$. If $L$ is a finite extension of $F$ then there is a homomorphism 
$\Cal P :\dfun(D \otimes_F L) \longrightarrow \dfun(D)$ such that 
$\Cal P \Cal I = \eta_{[L:F]}$, where $\eta_m(x)=x^m$. 

\demo{Proof}
Let $[L:F]=m$. Consider the regular representation  
$L \overset {\iota }\to \longrightarrow M_m(F)$ and the corresponding sequence
when we tensor over $F$ with $D$:

$$
D \longrightarrow D \otimes_F L  \overset {1\otimes \iota} \to 
\longrightarrow D \otimes_F M_m(F) \longrightarrow M_m(D)$$
$$a \longmapsto a \otimes 1 \longmapsto a \otimes 1 \longmapsto
aI_m \tag 2.1$$
$$1 \otimes \ell \longmapsto 1 \otimes \iota( \ell) \longmapsto 
\iota(\ell).$$
Since $\frak G$ is a Dieudonne functor, we obtain a homomorphism 
$\Cal P: \dfun(D \otimes_F L) \longrightarrow
\dfun(M_m(D)) \overset {d} \to \longrightarrow \dfun(D)$.
Again the sequence (2.1) shows that, thanks
to definition of a Dieudonne functor, $\Cal P \Cal I (x) = x^m$.
\qed
\enddemo

Note that in the above proposition $D$ could be an infinite 
dimensional division algebra. 
If $D$ is finite dimension over its 
center $F$, then it turns out that $\dfun(D)$, where $\dfun$
is reduced Dieudonne functor, is a torsion group.

\cor \label{dieutorsion}
Let $\dfun$ be a reduced Dieudonne functor. Then for any division algebra $D$ 
of index $n$, $\dfun(D)$ is a torsion 
group of bounded exponent $n^2=[D:Z(D)]$.

\demo{Proof}
Thanks to Proposition 2.5, for any finite field extension $L$ of $F=Z(D)$, we have
the sequence of homomorphisms $\dfun(D) \overset {\Cal I} \to \longrightarrow 
\dfun (D \otimes_F L) 
\overset{\Cal P} \to \longrightarrow \dfun (D)$, such that $\Cal P \Cal I(x)=
x^m$, where $x\in \dfun (D)$ and $[L:F]=m$.
Now let $L$ be a maximal subfield of $D$. Since $L$ 
is a splitting field for $D$, we get the sequence of homomorphisms 
$\dfun(D) \overset {\Cal I} \to 
\longrightarrow  \dfun (M_n(L)) \overset{\Cal P} \to \longrightarrow 
\dfun (D)$. Since $\dfun$ is a reduced Dieudonne functor, it follows that 
$\dfun (M_n(L))$ is a torsion group of bounded exponent $n$ (See  
Definition 2.2). Now the fact that for any $x \in \dfun (D)$,
$\Cal P \Cal I(x)=x^n$, shows that  $\dfun(D)$ is a torsion 
group of bounded exponent $n^2=[D:Z(D)]$.
\qed
\enddemo



Now it is immediate that if $A$ is a central simple algebra, then $\dfun(A)$
is also torsion.

We later specialize a Dieudonne functor and show that we can reduce this
bound for certain functors.
The following corollary  
shows that the analog result of 
the behavior of $SK_1(D)$ under extension of the ground field holds for 
a reduced Dieudonne functor. Namely,
we show that 
$\dfun(D)$ embeds in $\dfun(D \otimes_F L) $ when the index of $D$ and $[L:F]$
are relatively prime.

\cor \label{injection}
Let $D$ be a division ring over its center $F$ and $L/F$ be a finite field 
extension such that $[L:F]$ is relatively prime to the index of $D$. Then the 
canonical
homomorphism $\dfun(D) \overset {\Cal I} \to \longrightarrow \dfun(D
 \otimes_F L)$ is injective.

\demo{Proof}
Let $i(D)=n$ and $[L:F]=m$. Suppose $\Cal I (x)=1$ for some $x\in \dfun(D)$.
By Proposition 2.5, $\Cal P \Cal I(x)=x^m=1$. But $\dfun(D)$ is a torsion of 
bounded exponent $n^2$. Hence $x^{n^2}=1$. Since $m$ and $n$ are relatively 
prime, $x=1$ and the proof is complete. 
\qed
\enddemo

Computing a Dieudonne functor in general is a difficult task, as it 
covers groups of different nature like $SK_1(D)$, or the group 
$G(D)=D^*/F^*D'$.
In the next section we specialize a Dieudonne functor to $\dfun(D)=D^*/F^*D'$
and compute it for certain division algebras. But before we continue with
the functorial properties of $\dfun$, let us consider a class of reduced
Dieudonne functors which enjoy an additional property. Namely let
$\tau$ be a natural transformation $\tau : K_1 \longrightarrow \dfun$
such that,

(1) For any object $A$ in $\Cal C$, $\tau_A :K_1(A) \longrightarrow \dfun (A)$ 
is an epimorphism.

(2) For any division algebra $D$ and a nonnegative integer $n$, the following
diagram commutes,
$$\diagram
K_1(M_n(D)) \rto^{\tau} \dto^{det} & \dfun(M_n(D)) \dto^{d}\\
K_1(D) \rto^{\tau} & \dfun(D).
\enddiagram$$
For example, the functors of Example 2.3,
or $\dfun(D)=Nrd_{D/F}(D^*)/{F^*}^{i(D)}$ and $\dfun(A)=A^*/Nrd_{A/F}(A^*)A'$ 
enjoy the above property. 

Even in this case, it is not clear when a reduced Dieudonne functor is trivial. 
The following theorem is almost the only known example where
$\dfun(D)$ with above property is trivial.

\cor \label{trivial}
Let $D$ be a division ring of quaternions over real-closed field.
Let $\dfun$ be a reduced Dieudonne functor with a natural epimorphism
$\tau : K_1 \longrightarrow \dfun$ as above. Then $\dfun(D)=1$.


\demo{Proof}
For any finite field extension $L$ of $F=Z(D)$, the following
diagram is commutative,
$$\diagram
K_1(D \otimes_F L) \rto^{\quad \Cal P} \dto^{\tau} & K_1(D) \dto^{\tau}\\
\dfun(D \otimes_F L) \rto^{\quad \Cal P} & \dfun(D).
\enddiagram$$
Now since $D$ is algebraically closed (See [7], Section 16), thanks to 
Proposition 2.5 and above diagram,  
$\dfun(D \otimes_F L) \overset {\Cal P} \to \longrightarrow 
\dfun(D)$ is an  epimorphism. Replace $L$ by $\overline {F}$, the 
algebraic closure of $F$. 
Because $\overline {F}$ is a splitting field for $D$,  
$\dfun(D \otimes_F \overline {F}) = \dfun(M_2(\overline F))$. We show that 
$\dfun(M_2(\overline F))$ is a trivial group and hence the corollary follows. 
Since $\tau : K_1 \longrightarrow \dfun$ is a natural epimorphism,  
there is a sequence of homomorphism
$$\psi: K_1(\overline F)={\overline F}^* \overset{\simeq} \to \longrightarrow
K_1(M_2(\overline F)) \overset{epi.} \to \longrightarrow 
\dfun(M_2(\overline F)).$$
Take $x \in \dfun(M_2(\overline F))$. Since $\overline F$ is algebraically closed,
there exist $y \in K_1(\overline F)={\overline F}^*$ such that $\psi(y^2)=x$.
But $\dfun(M_2(\overline F))$ is a torsion group of bounded exponent 2, hence
$x=1$. This shows that  $\dfun(M_2(\overline F))$ is trivial and the proof is
complete.
\qed
\enddemo


Back to the functorial properties of $\dfun$, the next step is to replace 
the field $L$ in Proposition 2.5 by a 
division ring. The following proposition shows that the same 
result holds here too.

\prop \label{AtensorB} 
Let $\dfun$ be a Dieudonne functor. Let $A$ and $B$ be division rings with 
center $F$ such that $[B:F]$ is finite. Then there is a homomorphism 
$\Cal P :\dfun(A \otimes_F B) \longrightarrow \dfun(A)$ 
such that $\Cal P \Cal I = \eta_{[B:F]}$. 

\demo{Proof}    
Let $[B:F]=m$.
We have the following sequence of $F$-algebra homomorphisms,
$$ A \longrightarrow A \otimes_F B \longrightarrow A \otimes_F B \otimes_F B^{op} 
\longrightarrow A \otimes_F M_m(F) \longrightarrow M_m(A).$$
This implies the group homomorphism
$\Cal P: \dfun(A \otimes_F B) \longrightarrow
\dfun(M_m(A)) \overset {d} \to \longrightarrow \dfun(A)$. 
The rest of the proof is similar to one in 
Proposition 2.5.
\qed
\enddemo

Note that in the above proposition $A$ could be of infinite dimension over its 
center $F$. A same statement as Corollary 2.7 could be obtained here 
too. 
In particular if $(i(A):i(B))=1$ then
$\dfun(A)$ embeds in $\dfun(A \otimes_F B)$ and similarly for $B$.
Employing torsion theory of groups and sequences which are appeared in the 
above propositions, we can write the primary decomposition for $\dfun(D)$.
The proof follows more or less the same pattern as for $SK_1(D)$.

\thm
Let $\dfun$ be a reduced Dieudonne functor. 
Let $A$ and $B$ be division algebras with center $F$ such that
$(i(A), i(B))=1$. Then $\dfun(A \otimes_F B) = \dfun(A) \times \dfun(B)$.

\demo{Proof}
By Corollary 2.6, $\dfun(A \otimes_F B)$ is a torsion group of bounded 
exponent
$m^2n^2$ where $m=i(A)$ and $n=i(B)$. Therefore $\dfun(A \otimes_F B) \simeq
 \Cal G \times \Cal H$, where $exp(\Cal G) \vert m^2$ and $exp(\Cal H) 
\vert n^2$.
By Proposition 2.9, we have the sequence:
$$ \dfun(A) \overset \phi \to \longrightarrow \dfun(A \otimes_F B) \overset 
\psi \to
\longrightarrow \dfun(A \otimes B \otimes B^{op}) \overset \theta \to \longrightarrow
\dfun(A) \tag 2.2$$ such that $\theta \psi \phi = \eta_{n^2}$. Hence $\dfun(A)=
\eta_{n^2}
\eta_{n^2}(\dfun(A))=\eta_{n^2} \theta \psi \phi (\dfun(A)) \subseteq \theta \psi \eta_{n^2}
(\Cal G \times \Cal H)=\theta \psi (\Cal G) \subseteq \dfun(A)$. This shows that
$\theta \psi \vert_{\Cal G}: \Cal G \longrightarrow \dfun(A)$ is surjective. Next we show that $\theta \psi \vert_{\Cal G}$ is injective. Considering the regular
representation $B^{op} \longrightarrow M_{n^2}(F)$, by  
Proposition 2.5, we have the following sequence 
$$\dfun(A \otimes_F B) \overset \psi \to \longrightarrow \dfun(A \otimes B \otimes B^{op}) \overset \psi' \to \longrightarrow \dfun(A \otimes B \otimes M_{n^2}(F))
\overset \theta' \to \longrightarrow \dfun(A \otimes_F B)$$ such that
$\theta' \psi' \psi =\eta_{n^2}$. Now if $w \in \Cal G - {1}$, then 
$\theta' \psi' \psi (w) =\eta_{n^2}(w)=w^{n^2} \neq 1$. Therefore 
$\psi \vert_{\Cal G}$ is injective. 
Rewrite the sequence (2.2) as follows:
$$\dfun(A \otimes_F B) \overset \psi \to \longrightarrow \dfun(A \otimes B \otimes B^{op}) \overset {iso.} \to \longrightarrow \dfun(M_{n^2}(A)) 
\overset {d} \to \longrightarrow \dfun(A).$$ Suppose $x \in \Cal G$ such that 
$\theta \psi(x)=1$. The above
sequence and the definition of reduced Dieudonne functor shows that 
${\psi(x)}^{n^2}=1$. Since $\psi \vert_{\Cal G}$ is injective, $x^{n^2}=1$. 
On the other hand because $exp(\Cal G) \vert m^2$ then $x^{m^2}=1$. Since $m$ and $n$ are relatively prime, $x=1$. 
This shows that $\theta \psi$ is an isomorphism and so 
$\dfun(A) \simeq \Cal G$.
In the similar way it can be shown that $\dfun(B) \simeq \Cal H$. Therefore 
the proof is complete.
\qed
\enddemo


Now we specialize a Dieudonne functor to $\dfun(D)=D^*/F^*D'$ where $D$ is 
a division ring with center $F$. It turns out that the structure of this group carries 
significant information about the arithmetic of $D$.
Let us start with the following definition.


\define 
Let $A$ be a central simple algebra with center $F$.
For each integer $s \geq 0$ define $G_s(A)= A^*/{F^*}^sA'$ 
and $\displaystyle\inlimA = \lim_{\leftarrow_k} G_{s^k}(A)$ where 
$k \geq 1$.

Clearly $G_0(A)= K_1(A)$ and $G_1(A)= A^*/F^*A'$.
It is easy to see that $G_s(A)$ and $\inlimA$ are Dieudonne functors.
Also $\dfun(A)=G_1(A)$ is a reduced Dieudonne functor. 
It is immediate from Corollary 2.6 that $G_1(A)$ is a torsion group of bounded
exponent $m[D:Z(D)]$ where $A=M_m(D)$. So it follows that  $G_s(A)$ are all 
torsions.
Note that $\inlimA$ is not torsion in general.
As we mentioned above we specialize $\dfun$ to the group $G(A)=A^*/F^*A'$.
In this case we can reduce the bound of the group $G(A)$ as follows.

Let $D$ be a division algebra over its center $F$. If $a \in D$ is algebraic 
over $F$ of degree $m$, then by Wedderburn's factorization theorem, 
we can associate $m$ conjugates to $a$ such that the sum
and the product of them are in $F$. 
The first observation is used, for instance, in [2] to compute the center of the group 
$D^*/1+M_D$ (which we will use it in this note) where $M_D$ is the maximal 
ideal 
of the valuation ring of $D$ (See Section 3). 
The fact that the product of the conjugates is in $F$, shows that 
$a^{[F(a):F]}$ is in $F^*D'$. Therefore if $D$ is an algebraic division algebra
over its center F, then $G(D)= D^*/F^*D'$ is a torsion group.
This is used in [10] to investigate the role of $D'$ in the structure of a   
division ring.

Now let $D$ be division algebra over its center $F$ with index $n$. 
Let $N$ be a normal subgroup of $D^*$. For $x \in N$, it follows that 
$Nrd_{D/F}(x)$ is 
the product
of $n$ conjugates of $x$. This shows that $Nrd\vert_N: N \longrightarrow Z(N)$ is well
defined, where $Z(N) = F^* \cap N$ is the center of the group $N$.
We will use the following lemma in Section 3 for normal subgroups of $D^*$ 
which arise from a valuation on $D$.

\lemma \label{torsion}
Let $D$ be a division ring over its center $F$ with index $n$. Let $N$ be a
normal subgroup of $D^*$. Then $N^n \subseteq Z(N)[D^*,N]$.

\demo{Proof} Let $x \in N$. As stated above, 
$Nrd_{D/F}(x)=d_1x{d_1}^{-1} \cdots d_nx{d_n}^{-1}$. 
This implies that $x^n=Nrd_{D/F}(x)d_x$ where 
$d_x \in [D^*,N]$ and $Nrd_{D/F}(x) \in F^* \cap N$.
This shows that  $N^n \subseteq (F^* \cap N)[D^*,N]$.
\qed
\enddemo

If in the above lemma we take $N=D^*$, then for any $x\in D$,
$x^n=Nrd_{D/F}(x)d_x$ where $d_x \in D'$. This in effect shows 
that $G(D)=D^*/F^*D'$ is a torsion group of bounded exponent $n$.

In the next section we will show yet another $SK_1$-like property for
the group $G(D)$. Namely $G(D)$ satisfy the following stability,
$G(D) \simeq G\big(D((x))\big)$ where $G\big(D((x))\big)$ is the division ring
of Laurent series (Corollary 3.6).
We close this section by the following theorem, which shows that
the group $G(D)=D^*/F^*D'$ does not always follow the same pattern as the 
reduced Whitehead group $SK_1(D)$.

\thm {\bf (J. -P. Tignol)}
Let $D$ be a division algebra over its center $F$ with index $n$. Then the
following sequence where $\wp$ runs over the irreducible
monic polynomials of $F[x]$ and $n_\wp$ is the index of $D \otimes_F F[x]/\wp$,
is split exact.
$$ 1 \longrightarrow G(D) \longrightarrow G(D(x)) \longrightarrow
\bigoplus_\wp \dfrac{\Bbb Z}{n/n_{\wp} \Bbb Z} \longrightarrow 1.$$ 
\demo{Proof}
By Proposition 7 in [8], the sequence 
$$ 1 \longrightarrow K_1(D) \longrightarrow K_1(D(x)) \longrightarrow
\bigoplus_\wp n_\wp/n \Bbb Z \longrightarrow 1$$ which is
obtained from the localization exact sequence of algebraic $K$-theory
is split exact.
Now since the group $G(D)$ is the cokernel of the natural map
$K_1(F) \longrightarrow K_1(D)$, applying the snake lemma to the commutative 
diagram,
$$\diagram
1 \rto & K_1(F) \rto \dto & K_1(F(x)) \rto \dto
& \bigoplus_\wp \Bbb Z \rto \dto & 1 \\ 
1 \rto & K_1(D) \rto & K_1(D(x)) \rto &
\bigoplus_\wp n_\wp/n \Bbb Z \rto & 1 \enddiagram$$ the result follows.
\qed
\enddemo

\bigskip

\sec {On The Group $G(D)$ Over Henselian Division Algebra}

In this section we assume that $D$ is a finite dimensional division algebra
over a henselian field $F=Z(D)$. 
Recall that a valuation $v$ on $F$ is {\it Henselian} if and only if $v$ has a 
unique extension to each field algebraic over $F$. Therefore $v$ has a unique 
extension to $D$ (see [6] and [15]). 
Denote by $V_D, V_F$ the valuation rings,
$M_D, M_F$ their maximal ideals, and $\overline {D}, \overline {F}$ the residue division
 algebra and the residue field of $D$ and $F$ respectively. We also take 
$\Gamma_D, \Gamma_F$ for the value groups, $U_D, U_F$ for the groups of 
units of $V_D, V_F$ respectively.
Furthermore, we assume that $D$ is a {\it tame} division algebra, i.e., 
$Z(\overline D)$ is separable over $\overline F$ and $Char \overline F$ does 
not divide $i(D)$. 
The quotient group $\Gamma_D /\Gamma_F$ is called the {\it relative valued 
group} of the valuation.
In this setting it turns out that $D$ is {\it defectless}, namely we
have $[\overline D : \overline F][\Gamma_D :\Gamma_F]=[D:F]$. 
$D$ is said to be {\it unramified} over $F$ if $[\Gamma_D:\Gamma_F]=1$.
At the other extreme $D$ is said to be {\it totally ramified} if 
$[D:F]=[\Gamma_D:\Gamma_F]$. $D$ is called {\it semiramified} if 
$\overline D$ is a field and $[\overline D :\overline F]=
[\Gamma_D:\Gamma_F]=i(D)$. For a recent account of the theory of henselian
valued division algebras see [6].

We start with the following theorem which describes a fundemental connection 
between the group $G(D)$ and its residue version. 

\thm \label{exse}
Let $D$ be a tame division algebra over a henselian
field $F=Z(D)$ with index $n$. Let $L/F$ be a subfield of $D$. Then  
the following sequence is exact.
$$ 1 \longrightarrow {\overline D }^*/ {\overline L}^* \overline {D'}
\longrightarrow {D^*}/{L^*}D' 
\longrightarrow \Gamma_D/\Gamma_L \longrightarrow 1. \tag 3.1$$
\demo{Proof}
Consider the normal subgroup $1+M_D$ of $D^*$. Thanks to Lemma 2.12, 
we have $(1+M_D)^n \subseteq ((1+M_D) \cap F^*) [D^*,1+M_D]$.
But applying the Hensel lemma to the elements of the group $1+M_D$, shows 
that, this group is {\it n-divisible}. Therefore $1+M_D=(1+M_D)^n$.
Hence $1+M_D \subseteq (1+M_F)D'$. 
Now consider the reduction map $U_D \longrightarrow {\overline D}^*$.
We have the following sequence:
$$ \eqalign{ 
  {\overline D}^*
  \overset {\simeq} \to {\longrightarrow} U_D/1+M_D
  \overset {nat.}   \to  {\longrightarrow} U_D/(1+M_F)D'
 & \overset {nat.}  \to {\longrightarrow} U_D/(1+M_L)D'
   \overset {nat.}  \to  {\longrightarrow} \cr
 & \overset {nat.}  \to  {\longrightarrow} U_D/U_LD'
  \overset {\simeq} \to {\longrightarrow} L^*U_D/L^*D'.} $$
Therefore 
$ \psi: {\overline D }^*/ ({\overline L}^*) \overline {D'}
\longrightarrow {L^*}U_D/{L^*}D'$ is an isomorphism.
Considering the fact that $D^*/L^*U_D \simeq \Gamma_D/\Gamma_L$, the theorem
follows.
\qed
\enddemo

Now we are ready to compute $G(D)$ for some certain cases. The 
statements {\it i.} and {\it ii.} of the following theorem
were first appeared in [5] using results from reduced $K$-theory.


\thm \label{classify}
Let $D$ be a henselian division algebra tame over its center $F$ with index 
$n$. Then

{\it i.} If $D$ is unramified over $F$ then $G(D) \simeq G(\overline D)$.

{\it ii.} If $D$ is totally ramified over $F$ then $G(D)= \Gamma_D /\Gamma_F$.

{\it iii.} If $D$ is semiramified and 
$\overline D$ is cyclic over $\overline F$ then the following sequence 
where $N_{\overline D / \overline F}$ is the norm function, is exact. 
$$ 1 \longrightarrow N_{\overline D / \overline F}(\overline D)/ 
{\overline F}^n \longrightarrow G(D) \longrightarrow \Gamma_D / \Gamma_F 
\longrightarrow 1.$$

{\it iv.} If $D$ is unramified and $s$ is an integer relatively prime
to $Char(\overline F)$, then the sequence 
$1 \longrightarrow {\hat G}_s (\overline D) \longrightarrow {\hat G}_s (D)
\longrightarrow \hat\Gamma_{D_{(s)}}$ is exact.

\demo{Proof}
{\it i.} Writing \xrefn{exse} for $L=F$, we have:
 $$ 1 \longrightarrow {\overline D }^*/ {\overline F}^* \overline {D'}
\longrightarrow G(D) 
\longrightarrow \Gamma_D/\Gamma_F \longrightarrow 1.$$
Now if $(D,v)$ is unramified, namely $[\Gamma_D :\Gamma_F]=1$, then
${\overline D }^*/ {\overline F}^* \overline {D'} \simeq {D^*}/{F^*}D'$.
On the other hand $Z(\overline D)= \overline F$ and
$D^*=F^*U_D$. Therefore, for $a,b \in D^*$, the element $c=aba^{-1}b^{-1}$
may be written in the form $ c=
\alpha \beta {\alpha}^{-1} {\beta}^{-1}$, where $\alpha$  and $\beta \in U_D$.
This shows $\overline {D'}= {\overline D}'$, 
so $G(D) \simeq G(\overline D)$.
\bigskip
{\it ii.} If $D$ is totally ramified over $F$ then $\overline D =\overline F$.
Writing \xrefn{exse} for $L=F$, since the group 
${\overline D }^*/ {\overline F}^* \overline {D'}$ is trivial then
$G(D)= \Gamma_D /\Gamma_F$.
\bigskip
{\it iii.} Let $D$ be semiramified and 
$\overline D$ be cyclic over $\overline F$. Consider the norm 
function 
$N_{\overline D / \overline F}: {\overline D}^* \longrightarrow 
{\overline F}^*$.
Moreover for any $x\in U_D$ we have 
$\overline {Nrd_{D/F}(x)}=N_{\overline D / \overline F}(\overline x)$ 
(See [4]).
This shows that $\overline {D'} \subseteq Ker N_{\overline D / \overline F}$. But if
$x \in Ker N_{\overline D / \overline F}$ then by Hilbert theorem 90, there
is a $\overline a$ such that 
$ x =\overline a {\sigma(\overline a)}^{-1}$, where $\sigma$ is the generator
of $Gal(\overline D /\overline F)$. It is well known that the 
{\it fundemental homomorphism} $D^* \longrightarrow Gal(Z(\overline D)/ \overline F)$ is surjective. Therefore  
$\sigma:\overline D \longrightarrow \overline D$ is of the form 
$\sigma(\overline a)=\overline {cac^{-1}}$, for some $c \in D^*$.
This shows that $x \in \overline {D'}$.
Therefore $Ker N_{\overline D / \overline F}=\overline {D'}$.
Now it is easy to see that ${\overline D}^* / {\overline F}^* 
\overline {D'} 
\simeq N_{\overline D / \overline F}({\overline D}^*) / {{\overline F}^*}^n.$ 
So thanks to \xrefn{exse}, 
$ 1 \longrightarrow N_{\overline D / \overline F}({\overline D}^*)/ 
{{\overline F}^*}^n
\longrightarrow G(D) \longrightarrow \Gamma_D / \Gamma_F \longrightarrow 1$
is exact.
\bigskip
{\it iv.} We can write the similar statement as Theorem 3.1 for
$L=F^s$, where $s$ is an integer
relatively prime to $Char \overline F$.
In this case we only have to change $\Gamma_L$ to $s\Gamma_F$. Hence we obtain 
the exact sequence:
$$ 1 \longrightarrow {\overline D }^*/ ({\overline F}^*)^s \overline {D'}
\longrightarrow D^*/{F^*}^sD' 
\longrightarrow \Gamma_D /s\Gamma_F \longrightarrow 1.$$
Now because inverse limit is a left exact functor, we obtain the 
following exact sequence:
$$ 1 \longrightarrow  \lim_{\leftarrow_k} {\overline D }^*/ 
({\overline F}^*)^{s^k} 
\overline {D'} \longrightarrow \hat {G_s}(D) 
\longrightarrow  \lim_{\leftarrow_k} \Gamma_D / s^k \Gamma_F .$$
If $D$ is unramified, thanks to {\it i.} $\overline F = Z(\overline D)$ and
$\overline {D'}= {\overline D}'$. therefore the result follows.
\qed
\enddemo


\rem \label{cyclic}
If $\overline D$ is a cyclic field extension of $\overline F$, 
a modification of the proof of 
{\it iii.}
above shows that $N_{\overline D / \overline F}({\overline D}^*)/ 
{{\overline F}^*}^f
\longrightarrow {\overline D}^*/ {\overline F}^*  \overline {D'}$, where 
$[\overline D :\overline F]=f$, is always surjective. Therefore if 
$N_{\overline D / \overline F}({\overline D^*})/ {{\overline F}^*}^f =1$ then $G(D)=
\Gamma_D / \Gamma_F$.


\example
Let $\Bbb C$ be a field of complex numbers. Let $1\neq \sigma 
\in Gal( \Bbb C/ \Bbb R)$ where
$\Bbb R$ is real numbers. Then by Hilbert construction, $D= \Bbb C((x, \sigma))$ is a
division ring with center $F= \Bbb R((x^2))$. We show that $G(D)=\Bbb Z_2$.
$D$ has a natural valuation such that $\Gamma_D / \Gamma_F = 
\Bbb Z / 2 \Bbb Z = \Bbb Z_2$. 
Clearly $\overline D=\Bbb C$ and $\overline F=\Bbb R$.
Since $N_{\Bbb C/ \Bbb R}( \Bbb C)=\Bbb R^2$ by 
Theorem 3.2 {\it iii.}, 
$G(D)=\Gamma_D/\Gamma_F=\Bbb Z_2$.



There have been significant results on the structure of relative
valued group in the case of totally ramified algebra. Using Theorem 3.2
we can write interesting statements relate group structure of $G(D)$ to 
algebraic structure of $D$. Recall that the group $G(D)$ is torsion of 
bounded exponent $n$.

\thm \label{totalyramified}
Let $D$ be a valued division algebra tame and totally ramified over henselian 
field $F=Z(D)$ of index $n$. Then,

{\it i.} There is a one to one correspondence between isomorphism classes of $F$- subalgebra of $D$ and the subgroups of $G(D)$.

{\it ii.} $exp(G(D))$ divides the exponent of $D$, i.e., the order of $[D]$
in $Br(F)$, the Brauer group of $F$.

{\it iii.} $D$ is cyclic division algebra if and only if $exp(G(D))=n$.

\demo{Proof}
The theorem follows by comparing Theorem 3.2 {\it ii.}, with the
results on relative valued group in the case of totally ramified valuation
(See for example [15]).
\qed
\enddemo


\cor
Let $D$ be a finite dimensional division algebra over
its center $F$ such that $Char F \nmid i(D)s$. Then there is a 
monomorphism 
$\hat G_s(D) \longrightarrow \hat G_s(D((x)))$. In particular 
if  $Char F \nmid i(D)$ then $G(D) \backsimeq G(D((x)))$. 
\qed

Now we are in a position to use the group $G(D)$ to compute $SK_1(D)$.
The following theorem enables us to compute $SK_1(D)$ when, roughly speaking, 
$G(\overline D)$ is trivial. Note that we do not use any results from 
reduced $K$-theory.

\thm \label{sk1}
Let $D$ be a tame division algebra over henselian field $F=Z(D)$ of 
index $n$.

{\it i.} If ${\overline D }^*/ {\overline F}^* \overline {D'}=1$ then 
$SK_1(D)=\mu_n(F)/Z(D')$.

{\it ii.} If $D$ is a cyclic division algebra with a maximal cyclic 
extension $L/F$ such that
${\overline D }^*/ {\overline L}^* \overline {D'}=1$ then $SK_1(D)=1$.

\demo{Proof}
{\it i.} As the proof of Theorem 3.1 shows, we have a natural isomorphism,
$$ \psi: {\overline D }^*/ ({\overline F}^*) \overline {D'}
\longrightarrow U_D/U_FD'.$$ 
Now if ${\overline D }^*/ {\overline F}^* \overline {D'}=1$ then $U_D=U_FD'$. 
But $D^{(1)} \subseteq U_D$. This shows that
$D^{(1)}=\mu_n(F)D'$. Using the fact that $\mu_n(F) \cap D' =Z(D')$ the theorem
follows.
\bigskip
{\it ii.} The same proof as {\it i.} shows that if 
${\overline D }^*/ {\overline L}^* \overline {D'}=1$ then $U_D=U_LD'$. 
Therefore $D^{(1)} \subseteq U_LD'$. Let $x \in D^{(1)}$. Then $x=ld$ where
$l \in L$ and $d \in D'$. So $Nrd_{D/F}(x)=N_{L/F}(l)=1$. Hilbert theorem 90
for the cyclic extension $L/F$ guarantee that $l=a {\sigma(a)}^{-1}$, where 
$\sigma$ is a generator of $Gal (L/F)$. Now Skolem-Noether theorem implies that
$\sigma(a)=cac^{-1}$ where $c \in D^*$. Therefore $l=aca^{-1}c^{-1}$. 
This shows that $D^{(1)}=D'$.
\qed
\enddemo

% Note that if $D$ is tame and henselian over its center $F$ with index $n$, 
% the Hensel lemma can be used to show that $\mu_n(F)=\mu_n(\overline F)$.

The part {\it i.} of the above theorem shows that if $D$ is totally ramified, then 
$SK_1(D)=\mu_n(F)/Z(D')$.

We conclude both theorems of [9] which are obtained by using a heavy 
machinery of reduced $K$- theory, as natural examples of the above theorem.

\example
{\it For any division algebra $D$ with center 
$F= \Bbb R((x_1,\cdots,x_m))$ where
$\Bbb R$ is the real numbers, $SK_1(D)$ is trivial. }

\demo{Proof}
From number theory, 
it is well known that $[D:F]=2^s$ where $s \leq m$.
Since the complete field $F= \Bbb R((x_1,\cdots,x_m))$
has a natural valuation, then $D$ enjoys a valuation which is 
obviously tame. It is clear that $\overline F= \Bbb R$. Because 
the only division algebras over 
real numbers are either the quaternion $\Bbb H_{\Bbb R}$ or the field 
$\Bbb C$ of complex numbers,
therefore $\overline D= \Bbb H_{\Bbb R}$ or $\overline D= \Bbb C$.
Now Corollary 2.6 and Remark 3.3 show that in either case 
${\overline D }^*/ {\overline F}^* \overline {D'}=1$. Now by Theorem 3.7,  
$SK_1(D)\simeq \mu_{i(D)}(F)/Z(D')$. But clearly $\mu_{i(D)}(F)=\{ 1,-1 \}$ and 
because the index of $D$ is even, $-1 \in D'$. This implies that $SK_1(D)=1$.


\enddemo

\example
{\it For any division algebra with center $F= C((x_1,\cdots,x_m))$ where 
$C$ is an algebraically closed field, $SK_1(D)$ is cyclic.}

\example 
{\it Hilbert classical construction of division algebras.}
Let $L$ be a field and $\sigma \in Aut(L)$ such that $o(\sigma)=n$. 
Let $F=Fix(\sigma)$ be the fixed field of $\sigma$.
Hence $Gal(L/F)$ is a cyclic group with the generator $\sigma$. 
Let $D=L((x,\sigma))$ be the division ring of formal Laurent series.
It follows that $Z(D)=F((x^n))$ and $i(D)=n$. $D$ has a natural valuation, and it is easy 
to see that with this valuation $D$ is semiramified and $L((x^n))$ is a 
maximal subfield of $D$. 
Now by Theorem 2.7 {\it ii.}, $SK_1(D)$ is trivial.

\example From Theorem 3.7 {\it ii.}, it is immediate that reduced Whitehead group of division algebra over a local field is trivial.



Because most of the
interesting valued division algebras arise from the iterated formal power 
series fields, we may consider {\it $r$-iterated Henselian division 
algebras}.
Following Platonov in [13], we define inductively an $r$-iterated henselian field $F$ 
if its residue field $\overline F$ is an $(r-1)$-iterated henselian field.
\footnote{
See Ershov's comment in \cite{3} on iterated valued field.
Among other things, Considering iterated valued field, enables us to have 
more insight in each step of reduction.}
Let $(D_i, v_i), 0 \le i \le r-1$ be an $r$ iterated henselian 
division algebra ($\overline {D_{i}} = D_{i+1}$). 
Let $\Phi_i :U_{D_{i-1}} \longrightarrow {D_{i}}$ be the $i$-th 
natural reduction map.
Then $\Phi_i(\Phi_{i-1}(\cdots(\Phi_1(a))\cdots))$ is called $i$-iterated 
reduction, if it is defined. Denote the $r$ iterated henselian division
algebra by $D$, ($D=D_0, Z(D)=F=F_0$).
We also need the following notations in order to state the following lemma.
By $[U_D]_i$ and $[U_F]_i$ we denote the set of all elements of $D$ and $F$
respectively, such that $i$ iterated reduction defines. Also by  
$[1+M_D]_i$ and $[1+M_F]_i$, we denote the subsets of  $[U_D]_i$ and 
$[U_F]_i$ such that $i$ iterated reduction equals one. Clearly 
$[1+M_F]_1=1+M_F$. we can 
write the main lemma of [5] in this setting.

\lemma \label{mainlm}
Let $D$ be a $i$ iterated tame division algebra of finite
dimension over a henselian field $F=Z(D)$ with index $n$.

{\it i.} For each $a \in [1+M_D]_i$ there is $b \in [1+M_F]_i$ such that
$ab \in D'$.

{\it ii.} $[1+M_D]_i \subsetneqq [1+M_F]_iD'$.

\demo{Proof}
{\it i.} Let $a \in [1+M_D]_i$. Then $a$ is contained in a maximal subfield of 
$D$, say $L$.
Therefore $a \in [1+M_L]_i.$ By lemma 3 [13], we have 
$N_{L/F}([1+M_L]_i)=[1+M_F]_i$. So $Nrd_{D/F}(a)=N_{L/F}(a) \in [1+M_F]_i.$
Let $t=Nrd_{D/F}(a)$. Using inductive argument for Hensel lemma, we will show that
there exist $c\in [1+M_F]_i$ such that $c^n=t$.
Let $s \in 1+M_F=[1+M_F]_1$. Applying Hensel lemma for $f(x)=x^n-s$ gives 
$c \in 1+M_F$ such that $c^n=s$. Now it is not hard to see that 
$\Phi_1 ([1+M_F]_i)=[1+M_{\overline F}]_{i-1}$. Therefore 
$[1+M_F]_i/[1+M_F]_1 \simeq [1+M_{\overline F}]_{i-1}$. Now by induction, we 
conclude that $[1+M_F]_i$ is n-divisible. Therefore exist 
$c \in [1+M_F]_i$ such that $c^n=t$. Now $Nrd_{D/F}(a)=c^n$. So
$Nrd_{D/F}(ac^{-1})=1$. Hence
$ac^{-1} \in D^{(1)} \cap [1+M_D]_i$.
Applying the Platonov's generalized congruence theorem (cf. [13] and [4])
, we obtain
$ac^{-1} \in D'$. Take $b=c^{-1}$ and the proof is complete.
\bigskip
{\it ii.} Applying the first part of the lemma for $i=1$, in each step of
reduction we have, $1+M_{D_i} \subseteq (1+M_{K_i}){D_i}'$ where $K_i=Z(D_i)$.
First we  show that in each step of reduction, 
${D_i}' \nsubseteq 1+M_{D_i}$.
Consider the groups $\Delta = {D_i}^*/1+M_{D_i}$ and 
$P(D_i)= (1+M_{K_i}){D_i}'/(1+M_{D_i})$.
One can easily observe that $P(D_i) = \Delta '$ and  as Theorem 2.11
of [2] shows, the center of $\Delta$
is ${K_i}^*(1+M_{D_i})/(1+M_{D_i})$. We claim that $\Delta$ is not an abelian 
group,
for otherwise $U_{D_i} = U_{K_i} (1+M_{D_i})$ which implies that $D_i$ is 
totally
ramified. Thus $ \overline {{D_i}'} = \mu_e( \overline {K_i} )$, where
$e = exp(\Gamma_{D_i} / \Gamma_{K_i})$, (cf. the proof of Theorem 3.1 of
[15]) which leads us to a contradiction. Therefore ${D_i}'$ is not in 
$1+M_{D_i}$ and $\Delta$ is not 
abelian.
But $\Phi^{-1}(1+M_{D_i}) = [1+M_D]_{i+1}$. If
$D' \subseteq [1+M_D]_{i+1}$ then $\Phi(D') \subseteq \Phi([1+M_D]_{i+1})=
1+M_{D_i}$.
But ${D_i}' \subseteq \Phi(D')$ so  ${D_i}' \subseteq 1+M_{D_i}$ a 
contradiction.
\qed
\enddemo


\rem In the proof of {\it i.} above, we could use Lemma 2.12 and avoid the
Platonov congruence theorem.

\thm
Let $D$ be a tame division algebra of $r$ iterated over henselian field $F$ of
index $n$. If there is an $0 \leq \ell \leq r-1$ such that 
$\overline D_\ell / \overline F_\ell \overline {{D'}_\ell} = 1$ then 
$SK_1(D)\simeq \mu_n(F)/Z(D')$.

\demo{Proof}
For any $ 0 \leq k \leq r-1$, consider the $k+1-th$ reduction map 
$$[U_D]_{k+1} \overset{\Phi_{k+1}\Phi_k \cdots \Phi_1 }\to
\longrightarrow \overline{D_k}^*.$$ Thanks to Lemma 3.12 {\it i.}, we have:
$$ \overline {D_k}^* \overset{\simeq}\to \rightarrow 
[U_D]_{k+1}/ [1+M_D]_{k+1}
\overset {nat.}\to \rightarrow [U_D]_{k+1}/ [1+M_F]_{k+1}D' 
\overset {nat.}\to \rightarrow [U_D]_{k+1}/ [U_F]_{k+1}D'.$$ 
Therefore,
$$ \overline {D_k}^*/ \overline {F_k}^* \overline {D_k'} \overset {\simeq}\to 
\longrightarrow
[U_D]_{k+1}/ [U_F]_{k+1}D'.$$
Hence if there is a $\ell$ such that 
$\overline {D_\ell}^* / \overline {F_\ell}^* \overline {{{D'}_\ell}}^* = 1$ 
then 
$[U_F]_{\ell+1} D'= [U_D]_{\ell+1}$. By lemma 1 in [13] $D^{(1)} \subseteq 
[U_D]_{\ell+1}$ 
so  
$D^{(1)}= \mu_n(F)D'$. Using the fact that $\mu_n(F) \cap D'= Z(D')$ the 
theorem follows.
\qed
\enddemo 


Considering the fact that each henselian division algebra is 1 iterated
division algebra, we recover Theorem 3.7 from the above theorem.

\bigskip

\sec {On The Unitary Setting}

In this section we introduce the unitary version of the group $G(D)$ and obtain
the similar results in the unitary setting.
Let $D$ be a division ring with an involution $\tau$ over its center $F$ 
with index $n$. Let $S_\tau(D)= \{ a\in D \vert a^\tau=a \}$ be the subspace of
symmetric elements and $\Sigma_\tau (D)$ the subgroup of $D^*$ generated 
by nonzero
symmetric elements. Here we concentrate on the involution of the first kind,
i.e. $\Sigma_\tau (D) \cap F^* = F^*$.

\define 
Let $D$ be a division ring with an involution $\tau$. Then the group
$KU_1(D)= D^*/ \Sigma_\tau (D) D'$ is called unitary Whitehead group and the 
$GU(D)= \Sigma_\tau (D)D'/F^*D'$ the unitary version of $G(D)$.

We will prove that there is a stability theorem for $GU(D)$
similar to one in Corollary 3.6. The first part 
of the following theorem was first proved by Platonov and Yanchevskii [14].

\thm \label{univer}
Let D be a finite dimensional tame and unramified division algebra with an 
involution of the 
first kind over a henselian field Z(D)=F. Then 
$KU_1(\overline D) \backsimeq KU_1(D)$ and $GU(\overline D)\backsimeq GU(D)$ .

\demo{Proof}
Consider the following sequence:

$${\overline D}^* \longrightarrow U_D/1+M_D \longrightarrow F^*U_D/F^*(1+M_D)
\longrightarrow D^*/\Sigma_\tau (D)D'.$$
Because the valuation is unramified, 
we have $\Sigma_{\overline \tau}(\overline D)=\overline{\Sigma_\tau (D)\cap U_D}$
(See [14]), 
and ${\overline D}'= \overline{D'}$ (See Theorem 3.2 {\it i.}). Therefore we have 
the following isomorphism:
$ {\overline D}^*/ \Sigma_{\overline \tau} (\overline D) {\overline D}' \overset 
{\backsimeq} \to \longrightarrow  D^*/ \Sigma\tau (D)D'$.

For the second part, consider the following commutative diagram with exact 
rows:


$$\diagram
1 \rto & GU(\overline D) \rto \dto_{} & G(\overline D) \dto^{iso.} \rto & KU_1(\overline D) 
\dto^{iso.} \rto & 1
\\
1 \rto & GU(D) \rto & G(D) \rto & KU_1(D) \rto & 1.
\enddiagram$$

The two of the vertical arrows are isomorphisms, thanks to the first part 
of this theorem and Theorem 3.2 {\it i.}. Therefore the third one is also 
isomorphism which completes the proof.
\qed
\enddemo

If $D$ has an involution of the first kind, then $D((x))$ enjoys a natural 
involution which is induced by the one from $D$. Therefore if   
$Char F \nmid i(D)$ then thanks to the above theorem, we have 
$GU(D) \backsimeq GU(D((x)))$ which is a stability for $GU(D)$.

\remark{Acknowledgments} I thank Oleg Izhboldin for some
helpful conversations. I am also grateful to J. -P. Tignol for his attention 
to my work. 
\endremark


\Refs



\ref
\no 1
\by S. Akbari, M. Mahdavi-Hezavehi, M. G. Mahmudi
\paper Maximal subgroups of $GL_1(D)$ 
\jour J. Algebra
\yr 1999
\vol 217
\issue 2
\pages 422-433
\endref 







\ref
\no 2
\by M. Chacron
\paper c-valuations and normal c-ordering
\jour Canad. J. Math.
\yr 1989
\vol 41
\pages 14-67
\endref 


\ref 
\no 3
\by P. Draxl
\book  Skew Field
\publ London Math. Soc. Lecture Note Ser. Vol 81
\publaddr Cambridge, Univ. Press. Cambridge
\yr 1983
\endref


\ref
\no 4
\by Y. L. Ershov
\paper  Henselian valuation of division rings and the group $SK_1(D)$
\jour  Math. USSR Sb.
\vol 45
\yr 1983
\pages 63-71
\endref



\ref
\no 5
\by R. Hazrat, M. Mahdavi-Hezavehi, B. Mirzaii
\paper Reduced $K$-theory and the group $G(D)=D^*/F^*D'$
\jour Algebraic K-theory and 
its Application, H. Bass, A.O.Kuku, and C. Pedrini.(Ed).
World Sci. Publishing, River Edge, NJ.
\yr 1999 
\pages 403-409
\endref




\ref
\no 6
\by B. Jacob, A. R. Wadsworth 
\paper  Division algebras over Henselian fields
\jour  J. Algebra
\vol 128
\yr 1990
\issue 1
\pages 126-179
\endref



\ref 
\no 7
\by T. Y. Lam
\book  A first course in noncommutative rings  
\publ Springer-Verlag
\publaddr New York
\yr 1991
\endref





\ref
\no 8
\by D. W. Lewis, J. -P. Tignol
\paper  Square class groups and Witt rings of central simple algebras
\jour  J. Algebra
\vol 154
\issue 2
\yr 1993
\pages 360-376
\endref









\ref
\no 9
\by V. A. Lipnitskii
\paper  On the Tannaka-Artin problem over special fields
\jour  Soviet Math. Dokl.
\vol 17
\issue 3
\yr 1976
\pages 639-642
\endref







\ref
\no 10
\by M. Mahdavi-Hezavehi
\paper  On derived groups of division rings
\jour Comm. Algebra
\vol 23
\issue 3
\yr 1995
\pages 913-926
\endref


\ref 
\no 11
\by R. S. Pierce
\book  Associative Algebras  
\publ Springer-Verlag
\publaddr New York
\yr 1982
\endref








\ref
\no 12
\by V. P. Platonov
\paper  The Tannaka-Artin problem and reduced $K$-theory
\jour  Math USSR Izv.
\vol 10
\yr 1976
\pages 211-243
\endref







\ref
\no 13
\by V. P. Platonov
\paper  On reduced $K$-theory for $n$-fold Hensel fields
\jour  Soviet Math. Dokl.
\vol 20
\issue 6
\yr 1979
\pages 1436-1439
\endref




\ref
\no 14
\by V. P. Platonov, V. I. Yanchevskii
\paper Dieudonne's conjecture on the structure of unitary groups over
a division ring, and hermitian $K$-theory  
\jour  Math. USSR Izv.
\vol 25
\issue 3
\yr 1985
\pages 573-598
\endref



\ref
\no 15
\by J. -P. Tignol, A. R. Wadsworth
\paper Totally ramified valuations on finite-dimensional division algebras
\jour Tran. Amer. Math. Soc.
\vol 302
\issue 1
\yr 1987
\pages 223-250
\endref


\endRefs
\enddocument

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