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%% Markus Rost
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\newtheorem*{theorem}{Theorem}
\newtheorem*{Hilbert}{Hilbert's 90 for symbols}
\newtheorem*{corollary}{Corollary}
\theoremstyle{definition}
\newtheorem*{definition}{Definition}
\newtheorem*{example}{Example}
\newtheorem*{remark}{Remark}

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\title{Norm Varieties and Algebraic Cobordism}
\author{Markus Rost}
\address{Department of Mathematics,
  The Ohio State University,
  231 W 18th Avenue,
  Columbus, OH 43210, USA}
\email{rost@math.ohio-state.edu}
\urladdr{http://www.math.ohio-state.edu/\~{}rost}
%\date{\today}
\subjclass{12G05}
%\subjclass{55N22}
\keywords{Milnor's $K$-ring, Galois cohomology, Cobordism}

\begin{document}

\begin{abstract}
  We outline briefly results and examples related with the bijectivity
  of the norm residue homomorphism.  We define norm varieties and
  describe some constructions.  Further we discuss degree formulas
  which form a major tool to handle norm varieties.  Finally we
  formulate Hilbert's 90 for symbols which is the hard part of the
  bijectivity of the norm residue homomorphism, modulo a theorem of
  Voevodsky.
\end{abstract}

\maketitle

\section*{Introduction}

This text is a brief outline of results and examples related with the
bijectivity of the norm residue homomorphism---also called
``Bloch-Kato conjecture'' and, for the $\amod 2$ case, ``Milnor
conjecture''.

The starting point was a result of Voevodsky which he communicated in
1996.  Voevodsky's theorem basically reduces the Bloch-Kato conjecture
to the existence of norm varieties and to what I call Hilbert's 90 for
symbols.  Unfortunately there is no text available on Voevodsky's
theorem.

In this exposition $p$ is a prime, $k$ is a field with $\car k\neq p$
and $K^M_nk$ denotes Milnor's $n$-th $K$-group of~$k$
\cite{Milnor:70}, \cite{Rost:96}.

Elements in $K^M_nk/p$ of the form
\begin{displaymath}
  u=\{a_1,\dots,a_n\}\bmod p
\end{displaymath}
are called symbols ($\amod p$, of weight~$n$).

A field extension $F$ of~$k$ is called a splitting field of $u$ if
$u_F=0$ in $K^M_nF/p$.

Let
\begin{gather*}
  h_{(n,p)}\colon K^M_nk/p\to H^n_\et(k,\mu_p^{\otimes n})
  \\
  \{a_1,\dots,a_n\}\mapsto (a_1,\dots,a_n)
\end{gather*}
be the norm residue homomorphism.

\section{Norm Varieties}

All successful approaches to the Bloch-Kato conjecture consist of an
investigation of appropriate generic splitting varieties of symbols.
This goes back to the work of Merkurjev and Suslin on the case $n=2$
who studied the $K$-cohomology of Severi-Brauer varieties
\cite{Merkurjev-Suslin:82}.  Similarly, for the case $p=2$ (for $n=3$
by Merkurjev, Suslin \cite{Merkurjev-Suslin:86} and the author
\cite{Rost:1986x}, for all $n$ by Voevodsky \cite{Voevodsky:02}) one
considers certain quadrics associated with Pfister forms.  For a long
time it was not clear which sort of varieties one should consider for
arbitrary $n$,~$p$.  In some cases one knew candidates, but these were
non-smooth varieties and desingularizations appeared to be difficult
to handle.  Finally Voevodsky proposed a surprising characterization
of the necessary varieties.  It involves characteristic numbers and
yields a beautiful relation between symbols and cobordism theory.

\begin{definition}
  Let $u=\{a_1,\dots,a_n\}\bmod p$ be a symbol.  Assume that $u\neq
  0$.  A \emph{norm variety} for~$u$ is a smooth proper irreducible
  variety~$X$ over~$k$ such that
  \begin{enumerateA}
    \item
    \label{item:split}
    The function field~$k(X)$ of~$X$ splits $u$.
    \item
    $\dim X=d:=p^{n-1}-1$
    \item
    \label{item:sd}
    $\frac{s_d(X)}p\not\equiv 0\mod p$
  \end{enumerateA}
\end{definition}

Here $s_d(X)\in\LZ$ denotes the characteristic number of~$X$ given by
the $d$-th Newton polynomial in the Chern classes of~$TX$.  It is
known (by Milnor) that in dimensions $d=p^n-1$ the number~$s_d(X)$ is
$p$-divisible for any~$X$.  If $k\subset \LC$ one may rephrase
condition~\eqref{item:sd} by saying that $X(\LC)$ is indecomposable in
the complex cobordism ring $\amod p$.

We will observe in section~\ref{sec:degree-formulas} that the
conditions for a norm variety are birational invariant.

The name ``norm variety'' originates from some constructions of norm
varieties, see section~\ref{sec:exist-norm-vari}.

We conclude this section with the ``classical'' examples of norm
varieties.

\begin{example}
  The case $n=2$.  Assume that $k$ contains a primitive $p$-th
  root~$\zeta$ of unity.  For $a$, $b\in k^*$ let $A_\zeta(a,b)$ be
  the central simple $k$-algebra with presentation
  \begin{displaymath}
    A_\zeta(a,b)=\amida{u,v}{u^p=a,v^p=b,vu=\zeta uv}
  \end{displaymath}
  The Severi-Brauer variety $X(a,b)$ of~$A_\zeta(a,b)$ is a norm
  variety for the symbol $\{a,b\}\bmod p$.
\end{example}

\begin{example}
  The case $p=2$.  For $a_1$, \ldots, $a_n\in k^*$ one denotes by
  \begin{displaymath}
    \pfister{a_1,\ldots,a_n}=\bigotimes_1^n\qform{1,-a_i}
  \end{displaymath}
  the associated $n$-fold Pfister form \cite{Lam:73},
  \cite{Scharlau:85}.  The quadratic form
  \begin{displaymath}
    \varphi=\pfister{a_1,\ldots,a_{n-1}}\perp\qform {-a_n}
  \end{displaymath}
  is called a Pfister neighbor.  The projective quadric $Q(\varphi)$
  defined by $\varphi=0$ is a norm variety for the symbol
  $\{a_1,\ldots,a_{n}\}\bmod 2$.
\end{example}

\section{Degree formulas}
\label{sec:degree-formulas}
The theme of ``degree formulas'' goes back to Voevodsky's first text
on the Milnor conjecture (although he never formulated explicitly a
``formula'') \cite{Voevodsky:96}.  In this section we formulate the
degree formula for the characteristic numbers~$s_d$.  It shows the
birational invariance of the notion of norm varieties.

The first proof of this formula relied on Voevodsky's stable homotopy
theory of algebraic varieties.  Later we found a rather elementary
approach \cite{Merkurjev:2000x}, which is in spirit very close to
``elementary'' approaches to the complex cobordism ring
\cite{Quillen:71}, \cite{Buoncristiano-Hacon:83}.

For our approach to Hilbert's 90 for symbols we use also ``higher
degree formulas'' which again were first settled using Voevodsky's
stable homotopy theory~\cite{Borghesi:00}.  These follow meanwhile
also from the ``general degree formula'' proved by Morel and Levine
\cite{Levine-Morel:02} in characteristic~$0$ using factorization
theorems for birational maps \cite{MR1896232}.

We fix a prime $p$ and a number~$d$ of the form $d=p^{n}-1$.

For a proper variety~$X$ over~$k$ let
\begin{displaymath}
  I(X)=\deg\bigl(\Ch_0(X)\bigr)\subset\LZ
\end{displaymath}
be the image of the degree map on the group of $0$-cycles.  One has
$I(X)=i(X)\LZ$ where $i(X)$ is the ``index'' of~$X$, i.~e., the gcd of
the degrees $\degre{k(x)}k$ of the residue class field extensions of
the closed points~$x$ of~$X$.  If $X$ has a $k$-point (in particular
if $k$ is algebraically closed), then $I(X)=\LZ$.  The group~$I(X)$ is
a birational invariant of~$X$.  We put
\begin{displaymath}
  J(X)=I(X)+p\LZ
\end{displaymath}

Let $X$, $Y$ be irreducible smooth proper varieties over~$k$ with
$\dim Y=\dim X=d$ and let $f\colon Y\to X$ be a morphism.  Define
$\deg f$ as follows: If $\dim f(Y)<\dim X$, then $\deg f=0$.
Otherwise $\deg f\in \LN$ is the degree of the extension $k(Y)/k(X)$
of the function fields.

\begin{theorem}[Degree formula for $s_d$]
  \label{formula}
  \begin{displaymath}
    \frac{s_{d}(Y)}p=(\deg f)\frac{s_{d}(X)}p \mod J(X)
  \end{displaymath}
\end{theorem}

\begin{corollary}
  \label{birat}
  The class
  \begin{displaymath}
    \frac{s_{d}(X)}p\bmod J(X)\in \LZ/J(X)
  \end{displaymath}
  is a birational invariant.
\end{corollary}

\begin{remark}
  If $X$ has a $k$-rational point, then $J(X)=\LZ$ and the degree
  formula is empty.  The degree formula and the birational invariants
  $s_d(X)/p\bmod J(X)$ are phenomena which are interesting only over
  non-algebraically closed fields.  Over the complex numbers the only
  characteristic numbers which are birational invariant are the Todd
  numbers.
\end{remark}

We apply the degree formula to norm varieties.  Let $u$ be a
nontrivial symbol $\amod p$ and let $X$ be a norm variety for~$u$.
Since $k(X)$ splits~$u$, so does any residue class field~$k(x)$ for
$x\in X$.  As $u$ is of exponent~$p$, it follows that $J(X)=p\LZ$.

\begin{corollary}[Voevodsky]
  Let $u$ be a nontrivial symbol and let $X$ be a norm variety of~$u$.
  Let further $Y$ be a smooth proper irreducible variety with $\dim
  Y=\dim X$ and let $f\colon Y\to X$ be a morphism.  Then $Y$ is a
  norm variety for~$u$ if and only if $\deg f$ is prime to~$p$.
\end{corollary}

It follows in particular that the notion of norm variety is birational
invariant.  Therefore we may call any irreducible variety~$U$ (not
necessarily smooth or proper) a norm variety of a symbol~$u$ if~$U$ is
birational isomorphic to a smooth and proper norm variety of~$u$.

\section{Existence of norm varieties}
\label{sec:exist-norm-vari}

\begin{theorem}
  Norm varieties exists for every symbol~$u\in K_n^Mk/p$ for every~$p$
  and every~$n$.
\end{theorem}

As we have noted, for the case $n=2$ one can take appropriate
Severi-Brauer varieties (if $k$ contains the $p$-th roots of unity)
and for the case $p=2$ one can take appropriate quadrics.

In this exposition we describe a proof for the case $n=3$ using
fix-point theorems of Conner and Floyd in order to compute the
non-triviality of the characteristic numbers.  Our first proof for the
general case used also Conner-Floyd fix-point theory.  Later we found
two further methods which are comparatively simpler.  However the
Conner-Floyd fix-point theorem is still used in our approach to
Hilbert's~90 for symbols.

Let $u=\{a,b,c\}\bmod p$ with $a$, $b$,~$c\in k^*$.  Assume that $k$
contains a primitive $p$-th root~$\zeta$ of unity, let
$A=A_\zeta(a,b)$ and let
\begin{displaymath}
  \mathit{MS}(A,c)=\bmidb{x\in A}{\Nrd(x)=c}
\end{displaymath}
We call $\mathit{MS}(A,c)$ the Merkurjev-Suslin variety associated
with~$A$ and~$c$.  The symbol~$u$ is trivial if and only if
$\mathit{MS}(A,c)$ has a rational point \cite{Merkurjev-Suslin:82}.
The variety $\mathit{MS}(A,c)$ is a twisted form of $\SL(p)$.

\begin{theorem}
  Suppose $u\neq0$.  Then $\mathit{MS}(A,c)$ is a norm variety
  for~$u$.
\end{theorem}

Let us indicate a proof for a subfield $k\subset \LC$ (and for $p>2$).
Let $U=\mathit{MS}(A,c)$.  It is easy to see that $k(U)$ splits~$u$.
Moreover one has $\dim U=\dim A-1=p^2-1$.  It remains to show that
there exists a proper smooth completion~$X$ of~$U$ with nontrivial
characteristic number.

Let
\begin{displaymath}
  \bar U=\bmidb{[x,t]\in \LP(A\oplus k)}{\Nrd(x)=ct^p}
\end{displaymath}
be the naive completion of~$U$.  We let the group $G=\LZ/p\times
\LZ/p$ act on the algebra~$A$ via
\begin{displaymath}
  (r,s)\cdot u=\zeta^r u, \quad (r,s)\cdot v=\zeta^s v
\end{displaymath}
This action extends to an action on $\LP(A\oplus k)$ (with the trivial
action on~$k$) which induces a $G$-action on~$\bar U$.  Let $\Fix(\bar
U)$ be the fixed point scheme of this action.  One finds that
$\Fix(\bar U)$ consists just of the $p$ isolated points
$[1,\zeta^i\sqrt[p]c]$, $i=1$, \ldots,~$p$, which are all contained
in~$U$.

The variety~$U$ is smooth, but~$\bar U$ is not.  However, by
equivariant resolution of singularities \cite{Bierstone-Milman:97},
there exists a smooth proper $G$-variety~$X$ together with a
$G$-morphism $X\to \bar U$ which is a birational isomorphism and an
isomorphism over~$U$.  It remains to show that
\begin{displaymath}
  \frac{s_d(X)}p\not\equiv 0\mod p
\end{displaymath}
For this we may pass to topology and try to compute
$s_d\bigl(X(\LC)\bigr)$.  We note that for odd $p$, the Chern
number~$s_d$ is also a Pontryagin number and depends only on the
differentiable structure of the given variety.  Note further that $X$
has the same $G$-fixed points as~$\bar U$ since the desingularization
took place only outside~$U$.

Consider the variety
\begin{displaymath}
  \textstyle Z=\Bmidb{\left[\sum_{i,j=1}^px_{ij}u^iv^j,t\right]\in
  \LP(A\oplus k)}{\sum_{i,j=1}^px_{ij}^p=ct^p}
\end{displaymath}
This variety is a smooth hypersurface and it is easy to check
\begin{displaymath}
  \frac{s_d(Z)}p\not\equiv 0\mod p
\end{displaymath}
As a $G$-variety, the variety~$Z$ has the same fixed points as~$X$
(``same'' means that the collections of fix-points together with the
$G$-structure on the tangent spaces are isomorphic).  Let $M$ be the
differentiable manifold obtained from $X(\LC)$ and $-Z(\LC)$ by a
multi-fold connected sum along corresponding fixed points.  Then $M$
is a $G$-manifold without fixed points.  By the theory of Conner and
Floyd \cite{Conner-Floyd:64}, \cite{Floyd:71} applied to
$(\LZ/p)^2$-manifolds of dimension $d=p^2-1$ one has
\begin{displaymath}
  \frac{s_d(M)}p\equiv 0\mod p
\end{displaymath}
Thus
\begin{displaymath}
  \frac{s_d(X)}p\equiv\frac{s_d(Z)}p \mod p
\end{displaymath}
and the desired non-triviality is established.

\subsection*{The functions $\Phi_n$}
We conclude this section with examples of norm varieties for the
general case.

Let $a_1$, $a_2$, \ldots be a sequence of elements in~$k^*$.  We
define functions $\Phi_n=\Phi_{a_1,\ldots,a_n}$ in~$p^n$ variables
inductively as follows.
\begin{align*}
  \Phi_0(t)&=t^p
  \\
  \Phi_n(T_0,\ldots,T_{p-1})&=\Phi_{n-1}(T_0)\prod_{i=1}^{p-1}
  \bigl(1-a_n\Phi_{n-1}(T_i)\bigr)
\end{align*}
Here the~$T_i$ stand for tuples of $p^{n-1}$ variables.  Let
$U(a_1,\ldots,a_n)$ be the variety defined by
\begin{displaymath}
  \Phi_{a_1,\ldots,a_{n-1}}(T)=a_n
\end{displaymath}

\begin{theorem}
  Suppose that the symbol $u=\{a_1,\dots,a_n\}\bmod p$ is nontrivial.
  Then $U(a_1,\ldots,a_n)$ is a norm variety of~$u$.
\end{theorem}

\section{Hilbert's 90 for symbols}
\label{sec:hilberts-90-symbols}

The bijectivity of the norm residue homomorphisms has always been
considered as a sort of higher version of the classical Hilbert's
Theorem~90 (which establishes the bijectivity for $n=1$).  In fact,
there are various variants of the Bloch-Kato conjecture which are
obvious generalizations of Hilbert's Theorem~90:  The Hilbert's
Theorem~90 for $K_n^M$ of cyclic extensions or the vanishing of the
motivic cohomology group $H^{n+1}\bigl(k,\LZ(n)\bigr)$.  In this
section we describe a variant which on one hand is very elementary to
formulate and on the other hand is the really hard part of the
Bloch-Kato conjecture (modulo Voevodsky's theorem).

Let $u=\{a_1,\dots,a_n\}\in K^M_nk/p$ be a symbol.  Consider the norm
map
\begin{displaymath}
  \mathcal N_u=\sum_F N_{F/k}\colon \bigoplus_F K_1F\to K_1k
\end{displaymath}
where $F$ runs through the finite field extensions of~$k$ (contained
in some algebraic closure of~$k$) which split~$u$.  Hilbert's Theorem
90 for~$u$ states that $\ker \mathcal N_u$ is generated by the
``obvious'' elements.

To make this precise, we consider two types of basic relations between
the norm maps $N_{F/k}$.

Let $F_1$, $F_2$ be finite field extensions of~$k$.  Then the sequence
\begin{equation}
  \label{eq:1}
  K_1(F_1\otimes F_2) \xrightarrow{ (N_{F_1\otimes F_2/F_1} ,
  -N_{F_1\otimes F_2/F_2}) } K_1F_1\oplus K_1F_2 \xrightarrow{
  N_{F_1/k} + N_{F_2/k}}K_1k
\end{equation}
is a complex.

Further, if $K/k$ is of transcendence degree~$1$, then the sequence
\begin{equation}
  \label{eq:2}
  K_2 K\xrightarrow{d_K} \bigoplus_v K_1\kappa(v)\xrightarrow N K_1k
\end{equation}
is a complex.  Here $v$ runs through the valuations of $K/k$, $d_K$ is
given by the tame symbols at each~$v$ and $N$ is the sum of the norm
maps $N_{\kappa(v)/k}$.  The sum formula $N\circ d_K=0$ is also known
as Weil's formula.

We now restrict again to splitting fields of~$u$.  The maps
in~\eqref{eq:1} yield a map
\begin{displaymath}
  \mathcal R_u=\sum _{F_1,F_2} (N_{F_1\otimes F_2/F_1} ,
  -N_{F_1\otimes F_2/F_2})\colon \bigoplus_{F_1,F_2} K_1(F_1\otimes
  F_2) \to \bigoplus_F K_1F
\end{displaymath}
with $\mathcal N_u\circ \mathcal R_u=0$.  Let $C$ be the cokernel of
$\mathcal R_u$ and let $\mathcal N_u'\colon C\to K_1k$ be the map
induced by~$\mathcal N_u$.  Then the maps in \eqref{eq:2} yield a map
\begin{displaymath}
  \mathcal S_u=\sum _{K} d_K\colon \bigoplus_K K_2K \to C
\end{displaymath}
with $\mathcal N_u'\circ \mathcal S_u=0$ where $K$ runs through the
splitting fields of~$u$ of transcendence degree~$1$ over~$k$
(contained in some universal field).  Let $ H_0(u,K_1)$ be the
cokernel of $\mathcal S_u$ and let $N_u\colon H_0(u,K_1)\to K_1k$ be
the map induced by~$\mathcal N_u'$.

\begin{Hilbert}
  For every symbol~$u$ the norm map
  \begin{displaymath}
    N_u\colon H_0(u,K_1)\to K_1k
  \end{displaymath}
  is injective.
\end{Hilbert}

\begin{example}
  If $u=0$, then it is easy to see that $N_u$ is injective.  In fact,
  it is a trivial exercise to check that $\mathcal N_u'$ is injective.
\end{example}

\begin{example}
  The case $n=1$.  The splitting fields~$F$ of $u=\{a\}\bmod p$ are
  exactly the field extensions of~$k$ containing a $p$-th root of~$a$.
  It is an easy exercise to reduce the injectivity of~$N_u$ (in fact
  of $\mathcal N_u'$) to the classical Hilbert's Theorem~90, i.~e.,
  the exactness of
  \begin{displaymath}
    K_1L\xrightarrow{1-\sigma}K_1L\xrightarrow{N_{L/k}}K_1k
  \end{displaymath}
  for a cyclic extension $L/k$ of degree~$p$ with~$\sigma$ a generator
  of $\Gal(L/k)$.
\end{example}

\begin{example}
  The case $n=2$.  Assume that $k$ contains a primitive $p$-th
  root~$\zeta$ of unity.  The splitting fields~$F$ of $u=\{a,b\}\bmod
  p$ are exactly the splitting fields of the algebra~$A_\zeta(a,b)$.
  One can show that
  \begin{displaymath}
    H_0(u,K_1)=K_1A_\zeta(a,b)
  \end{displaymath}
  with $N_u$ corresponding to the reduced norm map~$\Nrd$
  \cite{Merkurjev-Suslin:92}.  Hence in this case Hilbert's 90 for~$u$
  reduces to the classical fact $\mathit{SK}_1A=0$ for central simple
  algebras of prime degree \cite{Draxl:83}.
\end{example}

\begin{example}
  The case $p=2$.  The splitting fields~$F$ of
  $u=\{a_1,\dots,a_n\}\bmod 2$ are exactly the field extensions of~$k$
  which split the Pfister form $\pfister{a_1,\ldots,a_n}$ or,
  equivalently, over which the Pfister neighbor
  $\pfister{a_1,\ldots,a_{n-1}}\perp\qform {-a_n}$ becomes isotropic.
  Hilbert's 90 for symbols $\amod 2$ had been first established
  in~\cite{Ro:88}.  This text considered similar norm maps associated
  with any quadratic form (which are not injective in general).  A
  treatment of the special case of Pfister forms is contained in
  \cite{Kahn:97}.
\end{example}

\begin{remark}
  One can show that the group $H_0(u,K_1)$ as defined above is also
  the quotient of $\oplus_F K_1F$ by the $R$-trivial elements in $\ker
  \mathcal N_u$.  This is quite analogous to the description of $K_1A$
  of a central simple algebra~$A$:  The group $K_1A$ is the quotient
  of $A^*$ by the subgroup of $R$-trivial elements in the kernel of
  $\Nrd\colon A^*\to F^*$.  Similarly for the case $p=2$:  In this
  case the injectivity of~$N_u$ is related with the fact that for
  Pfister neighbors~$\varphi$ the kernel of the spinor norm
  $\SO(\varphi)\to k^*/(k^*)^2$ is $R$-trivial.
\end{remark}

In our approach to Hilbert's 90 for symbols one needs a
parameterization of the splitting fields of symbols.

\begin{definition}
  Let $u=\{a_1,\dots,a_n\}\bmod p$ be a symbol.  A \emph{$p$-generic
  splitting variety} for~$u$ is a smooth variety~$X$ over~$k$ such
  that for every splitting field~$F$ of~$u$ there exists a finite
  extension $F'/F$ of degree prime to~$p$ and a morphism $\Spec F'\to
  X$.
\end{definition}

\begin{theorem}
  Suppose $\car k=0$.  Let $m\geq 3$ and suppose for $n\leq m$ and
  every symbol $u=\{a_1,\dots,a_n\}\bmod p$ over all fields over~$k$
  there exists a $p$-generic splitting variety for~$u$ of dimension
  $p^{n-1}-1$.  Then Hilbert's 90 holds for such symbols.
\end{theorem}

The proof of this theorem is outlined in \cite{Rost:98x}.

For $n=2$ one can take here the Severi-Brauer varieties and for $n=3$
the Merkurjev-Suslin varieties.  Hence we have:

\begin{corollary}
  Suppose $\car k=0$.  Then Hilbert's 90 holds for symbols of
  weight~$\leq 3$.
\end{corollary}

\relax

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  \providecommand{\WWWaddress}[1]{#1}
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\begin{thebibliography}{10}

\bibitem{MR1896232}
D.~Abramovich, K.~Karu, K.~Matsuki, and J.~W{\l}odarczyk, \emph{Torification
  and factorization of birational maps}, J. Amer. Math. Soc. \textbf{15}
  (2002), no.~3, 531--572 (electronic).

\bibitem{Bierstone-Milman:97}
E.~Bierstone and P.~D. Milman, \emph{Canonical desingularization in
  characteristic zero by blowing up the maximum strata of a local invariant},
  Invent. Math. \textbf{128} (1997), no.~2, 207--302.

\bibitem{Borghesi:00}
S.~Borghesi, \emph{Algebraic {Morava} {$K$}-theories and the higher degree
  formula}, thesis, 2000, Evanston,
  \WWWaddress{http://www.math.uiuc.edu/K-theory/0412}.

\bibitem{Buoncristiano-Hacon:83}
S.~Buoncristiano and D.~Hacon, \emph{The geometry of {C}hern numbers}, Ann. of
  Math. (2) \textbf{118} (1983), no.~1, 1--7.

\bibitem{Conner-Floyd:64}
P.~E. Conner and E.~E. Floyd, \emph{Differentiable periodic maps}, Academic
  Press Inc., Publishers, New York, 1964, Ergebnisse der Mathematik und ihrer
  Grenzgebiete, N. F., Band 33.

\bibitem{Draxl:83}
P.~K. Draxl, \emph{Skew fields}, London Mathematical Society Lecture Note
  Series, vol.~81, Cambridge University Press, Cambridge, 1983.

\bibitem{Floyd:71}
E.~E. Floyd, \emph{Actions of $({Z}\sb{p})\sp{k}$ without stationary points},
  Topology \textbf{10} (1971), 327--336.

\bibitem{Kahn:97}
B.~Kahn, \emph{La conjecture de {M}ilnor (d'apr\`es {V}. {V}oevodsky)},
  Ast\'erisque (1997), no.~245, Exp.\ No.\ 834, 5, 379--418, S\'eminaire
  Bourbaki, Vol.\ 1996/97.

\bibitem{Lam:73}
T.~Y. Lam, \emph{The algebraic theory of quadratic forms}, Benjamin/Cummings
  Publishing Co. Inc. Advanced Book Program, Reading, Mass., 1980, Revised
  second printing, Mathematics Lecture Note Series.

\bibitem{Levine-Morel:02}
M.~Levine and F.~Morel, \emph{Algebraic cobordism {I}}, preprint, 2002,
  \WWWaddress{http://www.math.uiuc.edu/K-theory/0547}.

\bibitem{Merkurjev:2000x}
A.~S. Merkurjev, \emph{Degree formula}, notes, May 2000,
  \hfill\break\WWWaddress
  {http://www.math.ohio-state.edu/\~{}rost/chain-lemma.html}.

\bibitem{Merkurjev-Suslin:82}
A.~S. Merkurjev and A.~A. Suslin, \emph{${K}$-cohomology of {S}everi\mhy
  {B}rauer varieties and the norm residue homomorphism}, Izv. Akad. Nauk SSSR
  Ser. Mat. \textbf{46} (1982), no.~5, 1011--1046, 1135--1136 (Russian),
  \transl{Math. USSR Izv. \textbf{21} (1983), 307--340}.

\bibitem{Merkurjev-Suslin:92}
\bysame, \emph{The group of ${K}\sb 1$-zero-cycles on {S}everi-{B}rauer
  varieties}, Nova J. Algebra Geom. \textbf{1} (1992), no.~3, 297--315.

\bibitem{Merkurjev-Suslin:86}
\bysame, \emph{Norm residue homomorphism of degree three}, Izv. Akad. Nauk SSSR
  Ser. Mat. \textbf{54} (1990), no.~2, 339--356 (Russian), \transl{Math. USSR
  Izv. \textbf{36} (1991), no.~2, 349--367}, also: LOMI-preprint (1986).

\bibitem{Milnor:70}
J.~Milnor, \emph{Algebraic ${K}$-theory and quadratic forms}, Invent. Math.
  \textbf{9} (1970), 318--344.

\bibitem{Quillen:71}
D.~Quillen, \emph{Elementary proofs of some results of cobordism theory using
  {S}teenrod operations}, Advances in Math. \textbf{7} (1971), 29--56 (1971).

\bibitem{Ro:88}
M.~Rost, \emph{On the spinor norm and {$A_0(X,K_1)$} for quadrics}, preprint,
  1988, \WWWaddress {http://www.math.ohio-state.edu/\~{}rost/spinor.html}.

\bibitem{Rost:1986x}
\bysame, \emph{{H}ilbert 90 for {$K_3$} for degree-two extensions}, Preprint,
  1986.

\bibitem{Rost:96}
\bysame, \emph{Chow groups with coefficients}, Doc. Math. \textbf{1} (1996),
  No. 16, 319--393 (electronic).

\bibitem{Rost:98x}
\bysame, \emph{Chain lemma for splitting fields of symbols}, preprint, 1998,
  \WWWaddress {http://www.math.ohio-state.edu/\~{}rost/chain-lemma.html}.

\bibitem{Scharlau:85}
W.~Scharlau, \emph{Quadratic and {H}ermitian forms}, Grundlehren der
  mathematischen Wissenschaften, vol. 270, Springer-Verlag, Berlin, 1985.

\bibitem{Voevodsky:96}
V.~Voevodsky, \emph{The {Milnor} conjecture}, preprint, 1996,
  Max-Planck-Institute for Mathematics, Bonn,
  \WWWaddress{http://www.math.uiuc.edu/K-theory/0170/}.

\bibitem{Voevodsky:02}
\bysame, \emph{On $2$-torsion in motivic cohomology}, preprint, 2001,
  \WWWaddress{http://www.math.uiuc.edu/K-theory/0502/}.

\end{thebibliography}

\end{document}

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