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\title{The basic correspondence of a splitting variety}
\author{Markus Rost}
\address{Fakult\"at f\"ur Mathematik,
Universit\"at Bielefeld,
Postfach 100131,
33501 Bielefeld,
Germany}
\email{rost \textit{at} math.uni-bielefeld.de}
\urladdr{http://www.math.uni-bielefeld.de/\char`\~rost}
%\curraddr{}
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\date{August 17, 2007}
%\date{June 8, 2005}

\begin{document}

\maketitle

\begin{center}
    Summary of the talk at the 2007 Abel Symposium, Oslo, Norway
\end{center}

\bigskip

We discussed some details of a construction used in the proof of the
generalized Milnor conjecture.  A general overview had been given by
Sasha Merkurjev in the morning.  Let me just repeat here the statement
of the conjecture:

Consider 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 $\car k\neq p$.

In the talk we concentrated on the ``basic correspondence'' of a
splitting variety.  The general reference for this
is~\cite{Rost/basic-corr}.  For norm varieties and their relation to
characteristic numbers and cobordism see~\cite{MR1957022}.

In~\cite{Rost/basic-corr} we introduced also the more abstract notion
of a ``special correspondence" on a variety (with the basic
correspondences on norm varieties as only known examples so far).
Varieties with special correspondences have been considered further
recently in \cite{LAG/267}.

The following text is an extended version
of~\cite{Rost/basic-corr-at07}.

The basic correspondence of a splitting variety of a symbol~$u$ (see
below) 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 fix a generator
of~$\mu_p$) 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 from~$\mu$ 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~$\rho$, the necessary
nontriviality condition reads as
\begin{equation}
  \label{eq:c-rho-neq-0}
  c(\rho)\neq 0
\end{equation}
where $$c(\rho)\in\LZ/p\LZ$$ is a certain integer $\amod p$
(see~\cite[Section~5, p.~13]{Rost/basic-corr} for the definition).  I
could verify condition~\eqref{eq:c-rho-neq-0} ``by hand'', so to
speak, namely by investigating the specific examples of norm varieties
I had constructed earlier in~\cite{Rost/chain-lemma}.

Once one knows~\eqref{eq:c-rho-neq-0}, it is surprisingly easy to
prove $p$-genericity of the norm variety using the functoriality of
the definition of~$\mu$ and~$\rho$.  The argument, essentially due to
Voevodsky, was discussed in the lecture.  It is described
in~\cite[Section~6]{Rost/basic-corr}.

The construction of the motive can be done in the general setting of
special correspondences, see~\cite[Section~7]{Rost/basic-corr}.  For
$p=2$ things become particularly easy,
see~\cite[Section~7.3]{Rost/basic-corr}.

For an illustration, let me describe the basic correspondence in 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.

%% \bibliography{MR,more}
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\raggedbottom \providecommand{\cprime}{$'$} \providecommand{\cyr}{}
  \providecommand{\bielefeld}{bie\-le\-feld}
  \providecommand{\Ktheory}{K-the\-o\-ry} \hyphenation{ar-chives}
\providecommand{\myurl}[1]{{\def\-{\discretionary{}{}{}}%
            \def~{\char`\~}\upshape$\langle$#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/~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/~rost/basic-corr.html}.

\bibitem{Rost/basic-corr-at07}
\bysame, \emph{Norm residue homomorphism}, Abstract for the talk at the
  Mathematische Arbeitstagung, Bonn, June 22-28, 2007,
  \myurl{www.math.uni-\bielefeld.de/~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/}.

\bibitem{LAG/267}
K.~Zainoulline, \emph{Special correspondences and {Chow} traces of
  {Landweber-Novikov} operations}, Preprint, 2007, Linear Algebraic Groups and
  Related Structures, Preprint Server, Bielefeld,
  \myurl{www.math.uni-\bielefeld.de/lag/man/267.html}.

\end{thebibliography}

\end{document}