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\title{A Note on $K$-theory of Azumaya algebras}



\begin{thanks}
{The first author acknowledges the support of EPSRC first grant
scheme EP/D03695X/1. Part of this work has been done in the Winter
of 2007 at KIAS, Seoul, Korea and the Summer of 2008 at ICTP,
Trieste, Italy. The authors would like to thank Raymond Hoobler for
several discussions regarding the subject.}
\end{thanks}

\author{Roozbeh Hazrat} \address{
Dept. of Pure Mathematics\\
Queen's University\\
Belfast BT7 1NN\\
United Kingdom}
% \curraddr{Mathematics Section, The Graduate Center,
%City University of New York, 365 Fifth Avenue, New York, NY
%10016-4309, USA}
 \email{r.hazrat@qub.ac.uk}

\author{Judith R. Millar}\address{
Dept. of Pure Mathematics\\
Queen's University\\
Belfast BT7 1NN\\
United Kingdom}



\email{jmillar12@qub.ac.uk}


\begin{document}


\begin{abstract}
For an Azumaya algebra $A$ which is free over its centre $R$, we
prove that $K$-theory of $A$ is isomorphic to $K$-theory of $R$ up
to its rank torsions. We conclude that $K_i(A,\Bbb Z/m)=K_i(R,\Bbb
Z/m)$ for any $m$ relatively prime to the rank and $i\geq 0$. This covers, for
example,  $K$-theory of division algebras, $K$-theory of Azumaya algebras over semi-local rings and
$K$-theory of graded central simple algebras indexed by a totally
ordered abelian group.
\end{abstract}




\maketitle

\section{introduction}

Let $R$ be a ring and $K_i(R)$, $i\geq 0$, be Quillen's $K$-groups.
The construction of $K$-groups is functorial.  Furthermore, $K_i$
functors induce identity maps on inner-automorphisms of a ring and
$K_i(R)\rightarrow K_i(M_tR)\rightarrow K_i(R)$ is multiplication by
$t$, where $R\rightarrow M_tR$ is the diagonal homomorphism,
$r\mapsto rI_t$.

For the class of division rings finite dimensional over their
centres (which are fields), Green et. al. \cite{green} proved that
$K$-theory of a division algebra is essentially the same as
$K$-theory of its centre, i.e., for a division algebra $D$ over its
centre $F$ of index $n$,  \begin{equation} \label{keq} K_i(D)
\otimes \mathbb Z[1/n] \cong K_i(F) \otimes \mathbb
Z[1/n].\end{equation}
 Their proof combines the above observations with the Skolem-Noether
theorem which guarantees that algebra homomorphisms in the setting
of central simple algebras are inner, and then uses the main result
of \cite{hal} which states that
$$\lim_{i\rightarrow \infty} M_{n^{2i}}F \cong \lim_{i\rightarrow
\infty}M_{n^{2(i+1)}}D.$$

This note is a continuation of \cite{azumayask1} where it was
observed that one can naturally deduce \eqref{keq} by using the fact
that an $F$-central simple algebra $A$ is a twisted form of a matrix
algebra, i.e., there is a finite field extension $L/F$ such that
$A\otimes_F L=M_k L$. However, using this takes us out of the
category of $F$-algebras to the category of field extensions of $F$.

Another equivalent defining property for a central simple algebra
$A$ is that the opposite algebra is the inverse element in the
Brauer group, i.e., $A\otimes_F A^{\text{op}}\cong M_m F$, where $m$
is the dimension of $A$ over $F$ as a vector space.  In this note we
will use this to prove \eqref{keq} type properties (see also
Remark~\ref{history}).  This enables us to extend our category from
that of central simple algebras to the category of Azumaya algebras
free over their centres. In particular we will prove that $K$-theory
of an Azumaya algebra over a semi-local ring is essentially the same
as $K$-theory of its centre. Along the same lines, another corollary
of this approach is to conclude that $K$-theory of graded central
simple algebras indexed by a totally ordered abelian group has
\eqref{keq} type property.  However, it remains as a question
whether the same can be said about any Azumaya algebra of constant
rank:

\begin{ques}\label{ques}
Let $A$ be an Azumaya algebra over its centre $R$ of constant rank
$n$. Then is it true that for any $i \geq 0$, $K_i(A) \otimes
\mathbb Z[1/n] \cong K_i(R) \otimes \mathbb Z[1/n]?$
\end{ques}

%\nnote{Here comes the remark about the different definitions of
%Azumaya algebras in the literature}




\begin{remark}\label{history}
The term Azumaya algebra originates from the work done by Azumaya in
his 1951 paper \cite{az}. The definition has developed since then,
and an Azumaya algebra is now defined to be an $R$-algebra $A$ such
that $A$ is faithfully projective as an $R$-module and the natural
homomorphism $A \otimes_R A^{\op} \ra \End_R(A)$ is an isomorphism
(see \cite{knus}, Chapter III, \S 5.1). An equivalent condition is
that (i) $A$ is a finitely generated $R$-module and (ii) for every
maximal ideal $\m$ of $R$, $A/\m A$ is a central simple algebra over
$R/\m$ (see \cite{knus}, Chapter III, Theorem 5.1.1). This
equivalence has been proven by Azumaya (\cite{az}, Theorem 15, page
130) under the additional assumption that $A$ is free over $R$, and
by Bass (\cite{bass}, Chapter III, Theorem 4.1) under the assumption
that $A$ is projective over $R$, thanks to the work of Auslander and
Goldman \cite{ausgold}. So for an Azumaya algebra $A$ as originally
defined by Azumaya with $A$ free over its centre $R$, our Theorem
\ref{mainth} covers its $K$-theory.


\end{remark}




\section{$\mathcal D$-functors}
In this section we set up an abstract functor, called a $\mathcal
D$-functor, which will handle several quotient groups coming from
$K$-groups. Some of the ideas behind this definition have already
been employed in \cite{sk12001} to define a functor in the category
of central simple algebras (compare this definition with the one in
\cite{sk12001} and \cite{azumayask1}).  One main distinction of this
definition, besides being defined over the category of Azumaya
algebras, is that the domain is restricted to a fixed base  ring.
The aim is to show that such functors have bounded torsion abelian
groups as their range.

%(For this functor one can also establish a decomposition property on the objects, but
%we don't pursue this here).

Let $\text{Az}(R)$ be the category of Azumaya algebras over the
commutative ring $R$, $C(R)$ the category of all commutative
$R$-algebras and $\mathcal Ab$ the category of abelian groups.
Consider a functor $\mathcal F:\text{Az}(R) \rightarrow \mathcal
Func(C(R),\mathcal Ab)$. Thus for any Azumaya $R$-algebra $A$,
$\mathcal F_A:C(R) \rightarrow \mathcal Ab$ is a functor from $C(R)$
to the category of abelian groups. Consider the following property
for $\mathcal F$:

(1) $\mathcal F_R$ is a trivial functor, i.e., $\mathcal F_R(S)=1$
for any $R$-algebra $S$.

\noindent Consider also the following properties at the base point
$R$:

(2) For any faithfully projective $R$-module $P$, there is a
homomorphism $$d: \mathcal F_{A \otimes \End_R(P)}(R) \rightarrow
\mathcal F_A(R)$$ such that the composition $$\mathcal F_A(R)
\rightarrow \mathcal F_{A \otimes \End_R(P)}(R) \rightarrow
\mathcal F_A(R)$$ is $\eta_{[P:R]}$ if $P$ has a constant rank
$[P:R]$. Here $\eta_k(x)=x^k$.

(3) If $P$ has a constant rank over $R$ then $\ker(d)$ is
$[P:R]$-torsion.


We call a functor $\mathcal F$ with the above three properties a
$\mathcal D$-functor with respect to the class of constant
faithfully projective modules.


\begin{remark}\label{simple}
When evaluating $\mathcal F$ at the base point, i.e., $\mathcal
F_A(R)$, one can drop $R$ and simply write $\mathcal F(A)$. With
this simplification, when $P$ is a free module of rank $n$, from the
condition (2) it follows that the composition of $\mathcal F(A)
\rightarrow \mathcal F(M_n A) \rightarrow \mathcal F(A)$ is
$\eta_n$.
\end{remark}

\begin{remark}
The goal is to show that $\mathcal F_A(R)$ is a torsion abelian
group. However, one would like to prove the same statement for any
$\mathcal F_A(S)$ where $S$ is an $R$-algebra. This could be done if
one further assumes:

(4) For any $S \in C(R)$, $\mathcal F_A(S)=\mathcal F_{A \otimes
S}(S)$.

\noindent Here $\mathcal F_{A \otimes S}$ is a functor from $C(S)$
to $\mathcal Ab$. There is no need to introduce an extra notation,
say, $\mathcal F_A^R$ and $\mathcal F_{A\otimes S}^S$ here. The
condition (4) simply states that evaluating $\mathcal F_A$ at $S$ is
the same as evaluating the functor $\mathcal F_{A \otimes S}$ at the
base point.
\end{remark}


\begin{theorem} \label{torsionlem} Let $A$ be an Azumaya algebra
over $R$ of constant rank $n$. Then $\mathcal F_A(R)$ is
$n^2$-torsion, where  $\mathcal F$ is a $\mathcal D$-functor.
\end{theorem}
\begin{proof}
Since we will be working with the functor $\mathcal F$ on the base
point, for an $R$-Azumaya algebra $A$, we simply write $\mathcal
F(A)$ instead of  $\mathcal F_A(R)$ (see Remark~\ref{simple} above).
The two $R$-algebra homomorphisms, $i:A \rightarrow A \otimes_R
A^{\text{op}}$ and $r:A^{\text{op}} \rightarrow
\End_R(A^{\text{op}})$ induce the homomorphisms $\mathcal F(A)
\rightarrow \mathcal F(A \otimes_R A^{\text{op}})$ and $\mathcal
F(A^{\text{op}}) \rightarrow \mathcal F(\End_R(A^{\text{op}}))$.
Since $A$ is  an Azumaya algebra over $R$, we have $A \otimes_R
A^{\text{op}}=\End_R(A)$ and $A$ is faithfully projective of rank
$n$ (see \cite{knus}, Chapter III, \S 5.1 and Theorem 5.1.1). By (2)
in the definition of a $\mathcal D$-functor, we have a homomorphism
$d:\mathcal F(\End_R(A)) \rightarrow  \mathcal F(R)$. But $\mathcal
F(R)$ is trivial by (1) and thus the kernel of $d$ is $\mathcal
F(\End_R(A))$ which is, by (3), $n$-torsion.  Consider the following
diagram
\begin{equation}
\begin{split}
\xymatrix{
& \mathcal F(A) \ar[d]^i \ar@/^/[ddr]^{\eta_n}&  \\
& \mathcal F(\End_R(A)) \cong  \mathcal F(A \otimes_R A^{\text{op}})
\ar[d]^r\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;&  \\
& \mathcal F(A \otimes_R \End_R(A^{\text{op}}))\ar[r]^-{d} & \mathcal F(A)}
\end{split}
\end{equation}

By (2), $d\circ r \circ  i=\eta_n$. But from the commutativity of
the diagram and the fact that $\mathcal F(\End_R(A))$ is
$n$-torsion, it follows that $\mathcal F(A)$ is $n^2$-torsion.
\end{proof}


For a ring $A$ with centre $R$, consider the inclusion $R\rightarrow
A$. This induces the map $K_i(R) \rightarrow K_i(A)$ for $i \geq 0$.
Consider the exact sequence
\begin{equation}\label{exse}
1 \rightarrow \ZK[i](A) \rightarrow K_i(R) \rightarrow K_i(A) \rightarrow \CK[i](A) \rightarrow 1
\end{equation}
where $\ZK[i]$ and $\CK[i]$ are the kernel and cokernel of the map
$K_i(R) \rightarrow K_i(A)$ respectively. It is not difficult to see
that one can consider $\CK[i]$ as the following functor:
\begin{align*}
\CK[i]: \text{Az}(R) &\rightarrow \mathcal Func(C(R),\mathcal Ab) \\
A &\mapsto \CK[i](A\otimes_R -).
\end{align*}
Thus for any morphism $S \rightarrow T$ in $C(R)$, we have
$\CK[i](A\otimes_R S) \rightarrow \CK[i](A\otimes_R T)$. We will
check that $\CK[i]$ (and in a similar manner $\ZK[i]$) is a
$\mathcal D$-functor. Condition (1) in the definition of a $\mathcal
D$-functor is straightforward to check. To prove the second
condition, since $A\otimes_R \End_R(P) \cong \End_A(P\otimes_R A)$
(see \cite{bouralgebra}, Chapter II, \S 5.3, Proposition 7) and
$P\otimes_R A$  is faithfully projective over $A$, it is enough to
work with faithfully projective modules over $A$.  Let $P$ be a
faithfully projective  $A$-(bi-)module.  Let $\mathcal P(A)$ and
$\mathcal P(\End_A(P))$ denote the categories of finitely generated
projective (left) modules over $A$ and $\End_A(P)$ respectively.



Consider the functors
\begin{align}\label{for}
\mathcal P(A) & \stackrel{\phi}{\longrightarrow} \mathcal P (\End_A(P)) \\
 X  & \longmapsto \End_A(P)\otimes_A X  \notag
\end{align}
and
\begin{align}\label{for1}
\, \, \, \,\,\, \,\,\,\,\, \, \, \, \,\,\,  \, \, \, \,\,\, \,\,\,\,\, \, \, \
\mathcal P (\End_A(P))   &\stackrel{\psi}{\longrightarrow} \mathcal P (A) \\
 Y  & \longmapsto \homm_A(P,A)\otimes_{\End_A(P)} Y \notag
\end{align}

By the Morita theory, $\psi$ is a natural equivalence (see
\cite{knus}, Chapter I, Theorem 9.1.3), thus it gives rise to the
isomorphism from $K_i(\End_A(P))$ to $K_i(A)$. Furthermore,
$\psi\phi(X)=\homm_A(P,X)$. Clearly if $P=A^k$, then
$\psi\phi(X)=X^k$, i.e., $k$ copies of $X$. Since $K_i$ are functors
which respect direct sums (\cite{Quillen}, \S2),  this composition
induces a multiplication by $k$ in the level of $K$-groups. This
implies that Condition (2) holds for $K$-functors {\it with respect
to free modules of finite ranks}.

If we could prove that $\ZK[i]$ and $\CK[i]$ are $\mathcal
D$-functors in the generality that we set up our definition of a
$\mathcal D$-functor, i.e., for all faithfully projective modules of
constant rank, then we would be able to answer Question~\ref{ques}
positively. However, at this stage, we can show that $\CK[i]$ and
$\ZK[i]$ are in fact $\mathcal D$-functors with respect to free
modules of finite ranks and, consequently, for
Theorem~\autoref{torsionlem} to follow, we need to have Azumaya
algebras free over their centres.  Examples of Azumaya algebras free
over their centres, apart from division algebras, are Azumaya
algebras over semi-local rings and graded central simple algebras.
Thus restricting to free modules, (\ref{for}) and (\ref{for1}) give
rise to the homomorphisms $\phi$ and $\psi$ on the level of
$K$-groups and we have the following commutative diagram,
\begin{equation}
\begin{split}
\xymatrix{
K_i(R)  \ar[r] \ar[d]^{=} & K_i(A)  \ar[r] \ar[d]^\phi & \CK[i](A) \ar[r] \ar[d]&1  \\
K_i(R)  \ar[r] \ar[d]^{\eta_k} & K_i(M_kA)  \ar[r] \ar[d]^\psi_\cong & \CK[i](M_kA) \ar[r] \ar[d]^d& 1 \\
K_i(R)  \ar[r]  & K_i(A)  \ar[r] & \CK[i](A) \ar[r]& 1 }
\end{split}
\end{equation}
where compositions of columns are $\eta_k$, and thus Condition (2)
holds. Now if $x \in \CK[i](M_kA)$ such that $d(x)=1$, then a
diagram chase shows that $x^k=1$, satisfying Condition (3) of a
$\mathcal D$-functor. In the same manner one can show that $\ZK[i]$
is a $\mathcal D$-functor with respect to free modules of finite
ranks.

We are now in  a position to prove that the $K$-theory of  Azumaya
algebras over semi-local rings and $K$-theory of  graded central
simple algebras are isomorphic to  $K$-theory of their centres up to
(their ranks) torsions.

\begin{theorem}\label{mainth}
Let $A$ be an Azumaya algebra free over its centre $R$ of rank $n$.
Then for any $i \geq 0$, $$K_i(A) \otimes \mathbb Z[1/n] \cong
K_i(R) \otimes \mathbb Z[1/n].$$
\end{theorem}
\begin{proof} The argument before the theorem shows that $\CK[i]$
(and in the same manner $\ZK[i]$) is a $\mathcal D$-functor with
respect to free modules of finite ranks, and thus by Theorem
\ref{torsionlem}, $\CK[i](A)$ and $\ZK[i](A)$ are $n^2$-torsion
abelian groups. Tensoring the exact sequence (\ref{exse}) by
$\mathbb Z[1/n]$, since $\CK[i](A)\otimes \mathbb Z[1/n]$ and
$\ZK[i](A)\otimes \mathbb Z[1/n]$ vanish, the result  follows.
\end{proof}


Recall that $K$-theory with coefficients is related to absolute
$K$-theory by the following exact sequence (see \cite{weibel})
\begin{equation}
1 \longrightarrow K_i(A) \otimes \Bbb Z[1/m] \longrightarrow
K_i(A,\Bbb Z/m) \longrightarrow {}_mK_{i-1}(A) \longrightarrow 1
\end{equation}
for any ring $A$ and $m\geq0$. The following corollary is now immediate.



\begin{cor}\label{modp}
Let $A$ be an Azumaya algebra free over its centre $R$ of rank $n$.
Then for any $m$ relatively prime to $n$, and for any $i\geq 0$,
$$K_i(A,\Bbb Z/m) \cong K_i(R,\Bbb Z/m).$$
\end{cor}
\begin{proof}

Consider the diagram
\begin{equation}
\begin{split}
\xymatrix{
1 \ar[r] & K_i(R) \otimes \Bbb Z[1/m]  \ar[r]\ar[d] &K_i(R,\Bbb Z/m)
\ar[r]\ar[d] &{}_mK_{i-1}(R) \ar[r]\ar[d]& 1\\
1 \ar[r] & K_i(A) \otimes \Bbb Z[1/m] \ar[r] &K_i(A,\Bbb Z/m) \ar[r] &{}_mK_{i-1}(A) \ar[r]& 1}
\end{split}
\end{equation}
Since $\ZK[i](A)$ and $\CK[i](A)$ are $n^2$-torsion and $n$ and $m$
are relatively prime, it is an easy exercise to show that the outer
vertical maps are isomorphisms. Thus the middle vertical map is an
isomorphism.
\end{proof}

\begin{cor}\label{semilocal}
Let $A$ be an Azumaya algebra over its centre $R$, a semi-local
ring, of rank $n$. Then for any $i \geq 0$, $K_i(A) \otimes \mathbb
Z[1/n] \cong K_i(R) \otimes \mathbb Z[1/n].$
\end{cor}
\begin{proof}
Since $A$ is finitely generated projective of constant rank and $R$
is a semi-local ring, it follows that $A$ is a free module over $R$
(see \cite{bourbaki}, Chapter II, \S 5.3, Proposition 5), and thus
the corollary follows from Theorem \ref{mainth}.
\end{proof}

Let us note that, for an $F$-central division algebra $D$, and
$i=1$, $\CK(D)$ is the group $D^*/F^*D'$ where $D^*$ is the
multiplicative group of $D$ and $D'$ is its  commutator subgroup.
This group has a direct application in solving the open problem
whether a multiplicative group of a division algebra has a maximal
subgroup \cite{hazwadmax}. Indeed, since $\CK(D)$ is torsion of
bounded exponent (by Theorem~\ref{torsionlem}), if it is not
trivial, it has maximal subgroups and therefore $D^*$ has (normal)
maximal subgroups. Thus finding the maximal subgroups in $D^*$
reduces to the case that $\CK(D)$ is trivial. It is a conjecture
that $\CK(D)$ is trivial if and only if $D$ is an ordinary
quaternion division algebra over a Pythagorean field. It was shown
in \cite{hazwadmax} that such division algebras do have (non-normal)
maximal subgroups. Thus if the above conjecture is settled
positively, one concludes that the multiplicative group of a
division algebra does have a maximal subgroup.


The theory of graded division algebras, indexed by a totally ordered
abelian group, has recently been given considerable attention as the
theory is closely related to the theory of division algebras
equipped with a valuation.  Let $D$ be a division algebra with a
valuation. To this one associates a graded division algebra $grD=
\bigoplus_{\gamma \in \Gamma_D}gD_\gamma$ where $\Gamma_D$ is the
value group of $D$ and $gD_\gamma$ comes from the principal
fractional ideals of the valuation ring (see \S4 in \cite{hwcor} for
details). As it is mentioned in \cite{hwcor}, even though
computations in the graded setting are easier (and discrete), it
seems not so much is lost in passage from $D$ to its corresponding
graded division algebra $grD$. This was a motivation to the
systematic study of graded central simple algebras and their
correspondences, notably by Boulagouaz \cite{boulag}, Hwang, Tignol
and Wadsworth \cite {hazwadsk1,hwalg,hwcor,tigwad} and to the
comparison of certain functors defined on these objects, notably the
Brauer group and the reduced Whitehead group.

Recall that a unital ring $R = \bigoplus_{ \gamma \in \Gamma}
R_{\gamma}$ is called a \emph{graded ring} if $\Gamma$ is a group,
each $R_{\gamma}$ is a subgroup of $(R, +)$ and $R_{\gamma} \cdot
R_{\delta} \subseteq R_{\gamma + \delta}$ for all $\gamma, \delta
\in \Gamma$. Throughout this paper, we will assume that $\Gamma$ is
a totally ordered abelian group (thus it is a torsion free abelian
group). The elements of $R_\gamma$ are called \emph{homogeneous of
degree $\gamma$}. A graded ring $E = \bigoplus_{ \gamma \in
\Gamma_{E}} E_{\gamma}$ is called a \emph{graded division ring} if
every non-zero homogeneous element has a multiplicative inverse.
Note that $\Gamma_{E}$, the support of $E$, is a group and  that
$E_{0}$ is a division ring. Since $\Gamma_{E}$ is totally ordered,
it follows that $E$ has no zero
divisors and that $E^*$, the multiplicative group of $E$, is  the
set $E^{h} \setminus \{0\}$. %Note that for a graded
%division ring $D$, the centre, $Z(D)$, is a graded field with
%grading $Z(D)= \bigoplus_{\ga \in \Ga_{D}} (Z(D) \cap D_{\ga})$.
Note that every graded $E$-module $M$ is a \emph{graded free}
$E$-module, i.e., $M$ is a graded $E$-module which is free as an
$E$-module with a homogeneous basis.  A \emph{graded field} is a
commutative graded division ring. A graded algebra $A$ over a graded
field $R$ is said to be a {\it graded central simple algebra} over
$R$ if $A$ is a simple graded ring (i.e., no non-trivial graded two
sided ideals), finite dimensional over $R$ with centre $R$.
\begin{cor}
Let $A$ be a graded central simple algebra over its graded centre
$R$ of degree $n$. Then  for any $i \geq 0$, $$K_i(A) \otimes
\mathbb Z[1/n] \cong K_i(R) \otimes \mathbb Z[1/n].$$
\end{cor}

\begin{proof}
By \cite{boulag}, Proposition 5.1 (see also \cite{hwcor},
Corollary~1.2), a graded central simple algebra $A$ over $R$ is an
Azumaya algebra. Since $R$ is a graded field (indexed by a torsion
free group), $A$ is a free $R$-module. The corollary now follows
from Theorem \ref{mainth}.
\end{proof}

The above result seems also to follow (in a hard way) by using
Corollary~\ref{semilocal}, in combination with the fundamental
theorem of $K$-theory, i.e.,  Theorem~8, its corollary and the
exercise on p. 38 of Quillen's paper \cite{Quillen}.




%\begin{theorem}
%Let $\mathcal F$ be a $\mathcal D$-functor. If $A$ and $B$ are
%Azumaya algebras with relatively prime ranks, then $\mathcal F_{A
%\otimes B}= \mathcal F_A \times \mathcal F_B$.
%\end{theorem}






\begin{thebibliography}{99}

%\bibitem{albert} A. Albert, Structure of Algebras, American Math.
%Soc., Colloquium Publ. 1961.

%\bibitem{bourbaki} N. Bourbaki, Commutative Algebra, Chapters 1-7,
%Springer Verlag 1989.

%\bibitem{draxl} P. Draxl,  Skew fields,
%LMS Lecture Note Series {\bf 81}, Cambridge University Press,
%Cambridge, 1983.

\bibitem{ausgold} M. Auslander, O. Goldman, {\em The Brauer group
of a commutative ring},  Trans. Amer. Math. Soc.,  {\bf 97} (1960),
367--409.


\bibitem{az} G. Azumaya, {\em On maximally central algebras},
Nagoya Math. J., {\bf 2} (1951), 119--150.


\bibitem{bass} H. Bass,  Lectures on topics in algebraic
$K$-theory, Notes by Amit Roy, Tata Institute of Fundamental
Research Lectures on Mathematics, No. 41, Tata Institute of
Fundamental Research, Bombay, 1967.




\bibitem{boulag} M. Boulagouaz, \paper{Le gradu\`e d'une alg\`ebre
\`a division valu\`ee}, Comm. Algebra, {\bf 23} (1995),  4275--4300.

\bibitem{bouralgebra} N. Bourbaki, Algebra, Chapters 1--3, Springer-Verlag, New York,
1989.

\bibitem{bourbaki} N. Bourbaki, Commutative Algebra, Chapters 1--7,
Springer-Verlag, New York, 1989.



 \bibitem{hal} B. Dawkins, I. Halperin, \paper{The isomorphism of
 certain cuntinuous rings}, Can. J. Math., {\bf18} (1966),  1333-1344.




\bibitem{green} S. Green, D. Handelman, P. Roberts,
\paper{$K$-theory of finite dimensional division algebras,} J. Pure
Appl. Algebra, {\bf 12} (1978), no. 2, 153--158.

\bibitem{sk12001} R. Hazrat, \paper{$\SK$-like functors for division
  algebras}, J. Algebra, {\bf 239} (2001), no. 2, 573--588.

\bibitem{azumayask1} R. Hazrat, \paper{Reduced $K$-theory of
Azumaya algebras}, J. Algebra, {\bf 305} (2006), 687--703.

\bibitem{hazmathz} R. Hazrat, \paper{$\SK$ of Azumaya algebras over  Hensel pairs}, Math. Z., To appear.


\bibitem{hazwadmax} R. Hazrat, A. R. Wadsworth, \paper{On maximal
subgroups of the multiplicative group of a division algebra}, J. Algebra, To appear.

\bibitem{hazwadsk1} R. Hazrat, A. R. Wadsworth, \paper{$\SK$ of graded division algebras,} in preparation.


\bibitem{hwalg} Y.-S. Hwang, A. R. Wadsworth, \emph{Algebraic extensions of graded and valued fields},
Comm. in Algebra, {\bf 27} (1999), 821--840.

\bibitem{hwcor} Y.-S. Hwang, A. R. Wadsworth, \emph{Correspondences
between valued division algebras and graded division algebras}, J.
Algebra, {\bf 220} (1999), 73--114.
%\bibitem{pierce} R. Pierce, Associative Algebras, Springer Verlag, 1982.


%\bibitem{jacwad} B. Jacob, A. Wadsworth, Division algebras
 % over Henselian fields, J. Algebra 128 (1990), no. 1, 126--179.

%\bibitem{lam} T.Y. Lam, A first course in noncommutative
%rings, Springer-Verlag, New York, 1991.



\bibitem{knus} M.-A. Knus, Quadratic and Hermitian forms over
  rings, Springer-Verlag, Berlin, 1991.



\bibitem{Quillen} D. Quillen, \emph{Higher algebraic $K$-theory I},
Lecture Notes in Mathematics 341, Springer Verlag, Berlin, 1973.



%\bibitem{plat} V.P. Platonov, \paper {The Tannaka-Artin problem and reduced
 % $K$-theory}, Math USSR Izv. {\bf 10} (1976) 211--243.


%\bibitem{raynaud} M. Raynaud, Anneaux locaux
%Hens\`eliens, LNM, {\bf 169}, Springer-Verlag, 1070.


%\bibitem{rosen} J. Rosenberg, Algebraic $K$-theory and its
%applications, GTM, 147. Springer-Verlag, 1994.



\bibitem{saltman} D. Saltman, Lectures on division
algebras, RC Series in Mathematics, AMS, no. 94, 1999.



\bibitem{tigwad} J.-P. Tignol, A. R. Wadsworth, \emph{Value functions
and associated graded rings for semisimple algebras}, Trans. Amer.
Math. Soc., To appear.



\bibitem{weibel} C. Weibel, \emph{Mayer-Vietoris sequences and Mod $p$ $K$-theory},
Lecture Notes in Mathematics 966, Springer Verlag, Berlin, 1982.


%\bibitem{strano1983} R. Strano, \paper{Principal homogenous spaces over Hensel rings},
%Proc. Amer. Math. Soc. {\bf 87}, No. 2, 1983, 208--212.


%\bibitem{suslin} A. Suslin, \paper{$\SK$ of division algebras and
 % Galois cohomology}, 75--99, Adv. Soviet Math.,
  %4, AMS, 1991.


%\bibitem{Vas} L. Vaserstein, \paper{On the Whitehead determinant for semilocal
%rings},  J. Algebra  {\bf 283}  (2005), 690--699.


%\bibitem{wadsval} A. Wadsworth, Valuation theory on finite dimensional
%division algebras, Fields Inst. Commun. 32, Amer. Math. Soc.,
%Providence, RI, (2002), 385--449.



\end{thebibliography}


\end{document}
