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\title{Norm Residue Homomorphism}

\author{Markus Rost\\
        Universit\"at Bielefeld\\
        Postfach 100131\\
        33501 Bielefeld\\
        Germany
}

\date{Mathematische Arbeitstagung June 22-28, 2007\\
      Max-Planck-Institut f\"ur Mathematik, Bonn, Germany}
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%\section{}

In the introductory part of the talk we formulated the norm residue
homomorphism
\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)\cup \dots \cup (a_n)
\end{gather*}
from Milnor's $K$-groups to Galois cohomology.  The \emph{generalized
Milnor conjecture} (aka Milnor-Bloch-Kato conjecture, aka \ldots)
states the bijectivity of this map for any prime~$p$, any $n$, and any
field~$k$ with $\car k\neq p$.

Further we discussed norm varieties and their relation to
characteristic numbers and cobordism.  See~\cite{MR1957022}.

Finally we considered what we call the ``basic correspondence of a
splitting variety''.  It is obtained by the following diagram, which
is essentially due to Voevodsky:
\begin{displaymath}
  \begin{CD}
    \llap{$u\in{}$}\ker \bigr[H^n_\et(k,\mu_p^{\otimes (n-1)})
    \longrightarrow H^n_\et(k(X),\mu_p^{\otimes (n-1)})\bigl]
    \\[7pt]
    @A\simeq Aj A
    \\[7pt]
    H_{\CM}^{n,n-1}(\CX,\LZ/p)
    \\[7pt]
    @VV\displaystyle \beta \circ Q_1\circ \cdots \circ Q_{n-2}V
    \\[7pt]
    \llap{$\mu\in{}$}H_{\CM}^{2b+1,b}(\CX,\LZ)
    \\[7pt]
    @VV\proj V
    \\[7pt]
    \hskip-10pt\text{homology of }\bigr[\Ch^b(X)\to \Ch^b(X^2) \to
    \Ch^b(X^3)\bigl]
    \\[7pt]
  \end{CD}
\end{displaymath}

\goodbreak

Here
\begin{displaymath}
  u=(a_1)\cup \dots \cup (a_n) \in H^n_\et(k,\mu_p^{\otimes n})
\end{displaymath}
is a symbol (we assume $\mu_p\subset k$) and $X$ is a smooth variety
over~$k$ over which the symbol is split, i.e., $u_{k(X)}=0$.

Furthermore, $\CX$ is the simplicial scheme
\begin{displaymath}
  \CX:\Lsimpliset X{X^2}{X^3}
\end{displaymath}
The map~$j$ relating motivic cohomology of~$\CX$ to Galois cohomology
is an isomorphism if one assumes the generalized Milnor conjecture in
weight~$n-1$.  For this one uses results from~\cite{MR1744945}.

Then one applies the Milnor operations~$Q_i$ in motivic cohomology
(these can be expressed in terms of the motivic Steenrod operations
similarly as in topology) and the Bockstein homomorphism~$\beta$.

One obtains the class
\begin{displaymath}
  \mu\in H_{\CM}^{2b+1,b}(\CX,\LZ),\qquad b=\frac{p^{n-1}-1}{p-1}
\end{displaymath}
which plays an essential role in Voevodsky's work on the generalized
Milnor conjecture, cf.~\cite{K-theory/0639}.  If $X$ is a norm variety
for the symbol~$u$, Voevodsky uses the class~$\mu$ to show that~$X$ is
a generic (up to extensions of degree prime to~$p$) splitting variety
for~$u$ and to split off from~$X$ a certain motive, the so-called
generalized Rost motive.  (For $p=2$ genericity and the construction
of the motive can be obtained in a much more elementary way using
quadratic forms.)  All this is essential for the final proof of the
conjecture (involving, as for $p=2$, Margolis homology and the
so-called ``injectivity'', settled in~\cite{Rost/chain-lemma}, see
also~\cite{MR2220090}).

An important step in handling~$\mu$ is to verify a certain
nontriviality condition.  Some ingredients for this part of
Voevodsky's work have not been written up in details yet, but it seems
that they will appear soon, cf.~\cite{K-theory/0844}.

Last year I was able to derive genericity and the construction of the
motive in a more ad hoc fashion, cf.~\cite{Rost/basic-corr}.  One
considers the standard spectral sequence for the simplicial
scheme~$\CX$ which leads to the map~$\proj$ as indicated in the
diagram.  Then one picks a representative
\begin{displaymath}
  \rho\in \Ch^b(X^2)
\end{displaymath}
of~$\proj(\mu)$.  I call any such element a \emph{basic correspondence
of the norm variety~$X$ of~$u$}.  Working with~$\rho$, I could verify
the necessary nontriviality condition ``by hand'', so to speak, namely
by investigating the specific examples of norm varieties I had
constructed earlier in~\cite{Rost/chain-lemma}.

For an illustration, let us look at the case $n=2$.  In this case
$b=1$.  For~$X$ we take a Severi-Brauer variety (of dimension~$p-1$).
Thus $\rho$ is an element in the Picard group of~$X^2$:
\begin{displaymath}
  \rho\in \Ch^1(X^2)=\Pic(X^2)
\end{displaymath}
If we pass to the algebraic closure~$\bar k$ of~$k$, then
\begin{displaymath}
  X_{\bar k}=\LP^{p-1}_{\bar k}
\end{displaymath}
and one finds
\begin{displaymath}
  \rho_{\bar k}=\pi_0^*[\CO(1)]- \pi_1^*[\CO(1)]\mod p\Pic(X^2_{\bar k})
\end{displaymath}
where
\begin{displaymath}
  \pi_0,\pi_1\colon X\times X\to X
\end{displaymath}
are the projections.

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%\begin{thebibliography}{999}
%\bibitem[X]{X} Author,F.: Title, Journal Number (Year) frompage-topage
%\end{thebibliography}

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\providecommand{\bielefeld}{bie\discretionary{}{}{}le\discretionary{}{}{}feld}
\providecommand{\Ktheory}{K-the\discretionary{}{}{}o\discretionary{}{}{}ry}
\providecommand{\myurl}[1]{$\langle$#1$\rangle$}
\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}

\begin{thebibliography}{1}

  \bibitem{Rost/chain-lemma}
  M.~Rost, \emph{Chain lemma for splitting fields of symbols},
  Preprint, 1998,
  \myurl{www.math.uni-\bielefeld.de/\char`~rost/chain-lemma.html}.

  \bibitem{MR1957022}
  \bysame, \emph{Norm varieties and algebraic cobordism}, Proceedings
  of the International Congress of Mathematicians, Vol. II (Beijing,
  2002) (Beijing), Higher Ed. Press, 2002, pp.~77--85.

  \bibitem{Rost/basic-corr}
  \bysame, \emph{On the basic correspondence of a splitting variety},
  Preprint, 2006,
  \myurl{www.math.uni-\bielefeld.de/\char`~rost/basic-corr.html}.

  \bibitem{MR2220090}
  A.~Suslin and S.~Joukhovitski, \emph{Norm varieties}, J. Pure
  Appl. Algebra \textbf{206} (2006), no.~1-2, 245--276.

  \bibitem{MR1744945}
  A.~Suslin and V.~Voevodsky, \emph{Bloch-{K}ato conjecture and
  motivic cohomology with finite coefficients}, The arithmetic and
  geometry of algebraic cycles (Banff, AB, 1998), NATO Sci. Ser. C
  Math. Phys. Sci., vol.  548, Kluwer Acad. Publ., Dordrecht, 2000,
  pp.~117--189.

  \bibitem{K-theory/0639}
  V.~Voevodsky, \emph{Motivic cohomology with {${\bf
  Z}/l$}-coefficients}, Preprint, 2003, K-theory Preprint Archives,
  No.~639, \myurl{www.math.uiuc.edu/\Ktheory/0639/}.

  \bibitem{K-theory/0844}
  C.~Weibel, \emph{Patching the norm residue isomorphism theorem},
  Preprint, 2007, K-theory Preprint Archives, No.~844,
  \myurl{www.math.uiuc.edu/\Ktheory/0844/}.
\end{thebibliography}

\end{document}