%% Copyright (C) 2006 by Markus Rost
%%
%% Created: September 5, 2006
%% Last Changes: September 11, 2006

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\title{On Galois Cohomology, Norm~Functions and Cycles}
\author{Markus Rost}
%\date{\today}
\date{Bielefeld, September 2006}

\begin{document}

\maketitle

\begin{slide}
  \textbf{\large Galois Cohomology}

  $p$ a prime
  \\
  $F$ a field, $\car F\neq p$

  $\bar F$ a separable closure of~$F$
  \\
  $G_F=\Gal(\bar F/F)$ the absolute Galois group

  \begin{gather*}
    H^n(F,\LZ/p)=H^n(G_F,\LZ/p)
    \\[30pt]
    H^n(F,\LZ/p)=\varinjlim_{L/F} H^n(\Gal(L/F),\LZ/p)
    \\[10pt]
    \text{\small (Limit over the finite Galois field extensions $L$ of
    $F$)}
  \end{gather*}

  $F^\times=F\setminus\{0\}$ the multiplicative group of~$F$

  $F$ contains a primitive $p\mkern1.5mu$-th root~$\zeta_p$ of unity
  \\
  $\mu_p\subset F^\times$ the subgroup generated by $\zeta_p$

  \begin{displaymath}
    H^n(F,\mu_p^{\otimes m})=H^n(F,\LZ/p)\otimes \mu_p^{\otimes m}
  \end{displaymath}
\end{slide}

\begin{slide}
  \textbf{Computation of ${H^1(F,\mu_p)}$:}
  \begin{displaymath}
    H^1(F,\mu_p)=\Hom(G_F,\mu_p)
  \end{displaymath}
  Hilbert Satz 90, Kummer theory:
  \begin{gather*}
    F^\times/(F^\times)^p\overset{\simeq}{\longrightarrow}
    H^1(F,\mu_p)
    \\[10pt]
    a \to (a) =[F(\sqrt[p]a)/F]
  \end{gather*}

  \bigskip \bigskip \bigskip \bigskip

  \textbf{Bloch-Kato conjecture:}
  \\[10pt]
  For any field $F$ with $\car F\neq p$, the Galois cohomology ring
  \begin{displaymath}
    \bigoplus_{n\geq 0} H^n(F,\mu_p^{\otimes n})
  \end{displaymath}
  is generated by ${H^1(F,\mu_p)}$
\end{slide}

\begin{slide}
  \textbf{The basic relation in $H^2(F,\mu_p^{\otimes 2})$:}
  \begin{displaymath}
    (a)\cup(1-a)=0 \qquad(a\in F\setminus\{0,1\})
  \end{displaymath}
  \textbf{Proof:}  $E=F(\alpha)$, $\alpha^p=a$
  \begin{align*}
    (a)\cup(1-a)&=(a)\cup (N_{E/F}(1-\alpha))
    \\
    &=N_{E/F}((a)_E\cup (1-\alpha))
    \\
    &=N_{E/F}((\alpha^p)\cup (1-\alpha))=0
  \end{align*}

  \textbf{Milnor's $K$-ring of a field $F$:}
  \begin{align*}
    K_*^{\rm M}F &= K_0F \oplus K_1F \oplus K_2F \oplus \cdots
    \\
    & = T_{\LZ} (F^\times) / \langle a \otimes (1-a), a\in F\setminus
    \{0,1\}\rangle
  \end{align*}
  \begin{alignat*}4
    K_0F & = \LZ &&\quad \text{(integers)}
    \\
    K_1F & = F^\times &&\quad \text{(multiplicative group)}
  \end{alignat*}
%% \begin{displaymath}
%%   K_2F =   (F^\times \otimes_Z F^\times ) /  \langle a \otimes (1-a), a\in
%%   k\setminus \{0,1\}\rangle
%% \end{displaymath}

  \textbf{Bloch-Kato conjecture:}
  \\[10pt]
  The ring homomorphism
  \begin{gather*}
    K_*^{\rm M}F/p\longrightarrow \bigoplus_{n\geq 0}
    H^n(F,\mu_p^{\otimes n})
    \\[10pt]
    a_1\otimes\cdots\otimes a_n \to (a_1)\cup \cdots \cup (a_n)
  \end{gather*}
  is bijective
\end{slide}

\begin{slide}
  The elements
  \begin{displaymath}
    (a_1)\cup \cdots \cup (a_n) \in H^n(F,\mu_p^{\otimes n})
  \end{displaymath}
  with $a_1,\ldots,a_n\in F^\times$ are called \textbf{symbols}

  \textbf{Bloch-Kato conjecture (mod $p$, weight~$n$):}
  \\[10pt]
  $H^n(F,\mu_p^{\otimes n})$ is additively generated by symbols

  \bigskip

  Proofs:
  \begin{align*}
    &n=1&&\text{classical, Hilbert's Satz 90}
    \\[10pt]
    &p=2,\, n=2&&\text{Merkurjev (1982)}
    \\[10pt]
    &n=2&&\text{Merkurjev/Suslin (1982)}
    \\[10pt]
    &p=2,\, n=3&&\text{Merkurjev/Suslin, Rost (1986)}
    \\[10pt]
    &p=2&&\text{Voevodsky (1996--2002)}
    \\[10pt]
    &\forall\, p, n ???&\hskip7.75pt&\text{Voevodsky/Rost (1997--2007
    ?)}
  \end{align*}
\end{slide}

\begin{slide}
  \textbf{$H^2(F,\mu_p)$ and the Brauer group}

  $\Br(F)=$ group of similarity classes of central simple algebras
  over~$F$

  $\Br(F)=$ set of isomorphism classes of skew fields with center~$F$
  (finite $F$-dimension)

  \textbf{Cyclic algebras:}  $\zeta_p\in F$, $a, b\in F^\times$
  \begin{displaymath}
    A(a,b)=\langle\, X,Y\mid X^p=a,\, Y^p=b,\, YX=\zeta_pXY\,\rangle
  \end{displaymath}

  There is a natural isomorphism
  \begin{displaymath}
    H^2(F,\mu_p) \overset{\simeq}{\longrightarrow} \pBr(F)
  \end{displaymath}
  $\pBr(F)=p\mkern1.5mu$-torsion subgroup of $\Br(F)$

  If $\mu_p\subset F$, symbols correspond to cyclic algebras:
  \begin{gather*}
    H^2(F,\mu_p^{\otimes 2}) \overset{\simeq}{\longrightarrow} \pBr(F)
    \\[10pt]
    (a)\cup(b) \to [A(a,b)]
  \end{gather*}
%% \begin{displaymath}
%% A(a,1-a)\simeq M_p(F)
%% \end{displaymath}
\end{slide}

\begin{slide}
  \textbf{The $H^3$-invariant for semisimple algebraic groups $G$}

  (Rost, Serre; 1993)

  $H^1(F,G)=$ isomorphism classes of principal homogeneous $G$-spaces
  over~$F$

  The $H^3$-invariant is a collection of maps
  \begin{displaymath}
    \Theta\colon H^1(F,G)\longrightarrow H^3(F,Q_G\otimes
    \mu_{N(G)}^{\otimes 2})
  \end{displaymath}
  functorial in~$F$ and~$G$

  $Q_G=$ Weyl invariant quadratic forms on the root lattice

  $Q_G=\LZ$ for simple~$G$
\end{slide}

\begin{slide}
  \textbf{Example:} $G=G_2$ ($\car F\neq 2$):
  \\[10pt]
  $H^1(F,G_2)=$ isomorphism classes of \textbf{octonion algebras}
  over~$F$
  \\
  The nontoral subgroup
  \begin{displaymath}
    (\LZ/2)^3 \overset{j}{\longrightarrow} G_2
  \end{displaymath}
  yields
  \begin{gather*}
    H^1(F,\LZ/2)^3 \overset{j}{\longrightarrow} H^1(F,G_2)
    \overset{\Theta}{\longrightarrow} H^3(F,\LZ/2)
%% \underset{\vphantom{j}}{}
    \\[10pt]
    ((a),(b),(c))\to [O(a,b,c)]\to (a)\cup(b)\cup(c)
  \end{gather*}
  \\
  \textbf{Example:} $G=F_4$ ($\car F\neq 3, \mu_3\subset F$):
  \\[10pt]
  $H^1(F,F_4)=$ isomorphism classes of \textbf{exceptional Jordan
  algebras} over~$F$
  \\
  The nontoral subgroup
  \begin{displaymath}
    (\LZ/3)^3 \overset{j}{\longrightarrow} F_4
  \end{displaymath}
  yields
  \begin{gather*}
    H^1(F,\mu_3)^3 \overset{j}{\longrightarrow} H^1(F,F_4)
    \overset{\Theta}{\longrightarrow} H^3(F,\LZ/3)
%% \underset{\vphantom{j}}{}
    \\[10pt]
    ((a),(b),(c))\to [J(a,b,c)]\to (a)\cup(b)\cup(c)
  \end{gather*}
\end{slide}

\begin{slide}
  \textbf{\large Multiplicative Norm Functions}

  Given a symbol
  \begin{displaymath}
    u=(a_1)\cup \cdots \cup (a_n) \in H^n(F,\mu_p^{\otimes n})
  \end{displaymath}

  Need some sort of multiplicative function~$\Phi$ in $p^n$ variables
  generalizing the classical examples:

  \textbf{Example:}  $n=2$: $\Phi$ is the reduced norm form
  \begin{displaymath}
    \Phi=\Nrd\colon A(a_1,a_2)\to F
  \end{displaymath}
  of the cyclic algebra corresponding to~$u$

  \textbf{Example:}  $n=3$, $p=2$:  $\Phi$ is the norm form of the
  octonion algebra $O(a_1,a_2,a_3)$

  \textbf{Example:}  $n=3$, $p=3$:  $\Phi$ is the norm form of the
  exceptional Jordan algebra $J(a_1,a_2,a_3)$

  \textbf{Example:}  $p=2$:  $\Phi$ is the Pfister quadratic form
  $\Phi= \pfister{a_1,\ldots,a_n}$
\end{slide}

\begin{slide}
  Recall from (complex) cobordism:
  \\[10pt]
  $s_d(X)\in\LZ$ is Milnor's characteristic number
  \\[10pt]
  If $d=\dim X=p^m-1$, then $s_d(X)\in p\LZ$
%% $\frac{s_d(X)}{p}\in \LZ$

  Using algebraic cobordism and degree formulas one shows:

  \textbf{Theorem:}  If $u\neq 0$, there exists a rational function
  \begin{displaymath}
    \Phi\colon A\longrightarrow \LA^1
  \end{displaymath}
  on some variety~$A$ such that:
  \begin{itemizeS}
    \item
    $(u)_{F(A)}\cup (\Phi)=0$ in $H^{n+1}(F(A),\mu_p^{\otimes (n+1)})$
    \item
    $\dim A=p^n$
    \item
    For any smooth compactification~$X$ of the generic fiber of~$\Phi$
    one has
    \begin{displaymath}
      \frac{s_d(X)}{p}\neq0 \mod p
    \end{displaymath}
  \end{itemizeS}
  $(A,\Phi)$ is unique ``up to extensions of degree prime to~$p$'' (at
  least for $n=2$ or $p=2$)
\end{slide}

\begin{slide}
  \textbf{Multiplicativity of~$\Phi$:}  Ideally this means
  \begin{displaymath}
    \Phi(\mu(x,y))=\Phi(x) \Phi(y)
  \end{displaymath}
  for some (bilinear, rational?) map
  \begin{displaymath}
    \mu\colon A\times A \longrightarrow A
  \end{displaymath}

  Look for a correspondence
  \begin{displaymath}
    \mu\colon A\times A \overset{f}{\longleftarrow} W
    \overset{g}{\longrightarrow} A
%%
    \underset{\vphantom{f}}{}
  \end{displaymath}
  with $(\deg f,p)=1$

  This involves:
  \begin{itemizeS}
    \item
    Existence of \emph{generic} splitting varieties of symbols
    (Voevodsky, see next pages)
    \item
    Algebraic cobordism (Morel/Levine)
    \item
    Parameterization of the ``subfields'' of the ``algebra $A$ with
    norm~$\Phi$''---motivated by chain lemma for exceptional Jordan
    algebras (Serre, Petersson/Racine 1995)
  \end{itemizeS}
\end{slide}

\begin{slide}
  \textbf{\large Construction of certain Cycles}

  $u=(a_1)\cup \cdots \cup (a_n) \in H^n(F,\mu_p^{\otimes n})$ a
  symbol

  $X$ a splitting variety of~$u$: $u_{F(X)}=0$

  Using Bloch-Kato conjecture in weight $n-1$, get an element
  \begin{displaymath}
%% b=1+p+p^2+\cdots +p^{n-2}
    \eta_u\in \Ch^b(X^2)\qquad b=\frac{p^{n-1}-1}{p-1}
  \end{displaymath}
  in the Chow group of $b\mkern1.5mu$-codimensional cycles

  \textbf{Example:}  $n=2$:  $X=$ Severi-Brauer variety of cyclic
  algebra $A(a_1,a_2)$
  \begin{displaymath}
    (\eta_u)^{p-1} = \text{Diagonal}(X) + \text{decomp.~elements}
  \end{displaymath}

  \textbf{Example:}  $p=2$:  $X=$ Quadric with quadratic form
  $\pfister{a_1,\ldots,a_{n-1}}\perp\qform{-a_n}$
  \begin{displaymath}
    \eta_u = \text{``Rost projector''} + \text{decomp.~elements}
  \end{displaymath}
\end{slide}

\begin{slide}
  $H_{\CM}^{r,s}$: motivic cohomology (Suslin, Voevodsky)

  $\CX=$ simplicial scheme :  $\Lsimpliset X{X^2}{X^3}$

  $\beta=$ Bockstein
  \\
  $Q_i$ Steenrod/Milnor operations (Voevodsky)

  The map $j$ is an isomorphism assuming the Bloch-Kato conjecture in
  weight $n-1$

  \textbf{Construction of $\eta_u$:}
%%
  \bigskip
  \begin{displaymath}
    \begin{CD}
      \llap{$u\in{}$}\ker [H^n(F,\mu_p^{\otimes (n-1)})
      \longrightarrow H^n(F(X),\mu_p^{\otimes (n-1)})]
      \\[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]
      H_{\CM}^{2b+1,b}(\CX,\LZ)
      \\[7pt]
      @VV\rm projV
      \\[7pt]
      \hskip-10pt\text{Homology of }[\Ch^b(X)\to \Ch^b(X^2) \to
      \Ch^b(X^3)]
    \end{CD}
  \end{displaymath}
\end{slide}

\begin{slide}
  \textbf{Problem:}  Find some variety~$X$ such that:
  \begin{enumerateS}
    \item
    $(u)_{F(X)}=0$ in $H^n(F(X),\mu_p^{\otimes n})$
    \item
    $d=\dim X=p^{n-1}-1$
    \item
    The integer
    \begin{displaymath}
      c(X)=(\pi_1)_*(\eta_u^{p-1}) \in\Ch^0(X)=\LZ
    \end{displaymath}
    is nonzero mod~$p$
  \end{enumerateS}

  Then $X$ would be a generic splitting variety (up to extensions of
  degree prime to~$p$)

  \textbf{Theorem:}  There exists $X$ with (1), (2) and
  \begin{displaymath}
    \frac{s_d(X)}{p}\neq 0\mod p
  \end{displaymath}

  Voevodsky announced essentially that
  \begin{displaymath}
    c(X)=\frac{s_d(X)}{p}\mod p
  \end{displaymath}
\end{slide}

\end{document}
