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\begin{document}

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\section*{Milnor K-theory}

\emph{The following text is copied from \href{http://mathoverflow.net/questions/4246/why-is-milnor-k-theory-not-ad-hoc/4258#4258}{MO/4246/4258} by [[Denis-Charles Cisinski]]:}

\textbf{Milnor K-theory} gives a way to compute [[étale cohomology]] of [[fields]] (i.e. [[Galois cohomology]]): if E is a field of characteristic different from a prime l, there is a residue map from the nth Milnor K-group of E mod l to the nth \'e{}tale cohomology group of E with coefficients in the sheaf of lth [[roots of unity]] to the n (i.e. tensored with itself n times). There is the [[Bloch-Kato conjecture]], which predicts that these residue maps are bijective. It happens that the case l=2 was conjectured by [[John Milnor|Milnor]] (up to a reformulation I guess). The [[Milnor conjecture]] has been proved by [[Vladimir Voevodsky|Voevodsky]] (and it was the first great achievements of [[homotopy theory of schemes]], which he initiated with [[Fabien Morel|Morel]] during the 90's), and he got his Fields medal in 2002 for this. Now [[Markus Rost|Rost]] and Voevodsky claimed they have a proof of the full Bloch-Kato conjecture for any prime l (which should appear some day, thanks to the work of quite a few people, among which [[Charles Weibel]] is not the least). Note also that the Bloch-Kato conjecture makes sense for l=p=char(E), but then, you have to replace \'e{}tale cohomology by [[de Rham-Witt cohomology]] (and this has also been proved by Bloch and Kato). [[Andrei Suslin|Suslin]] and Voevodsky also proved that the Bloch-Kato conjecture implies the [[Beilinson-Lichtenbaum conjecture]], which predicts the precise relationship between torsion [[motivic cohomology]] of varieties with torsion [[étale cohomology]].

Milnor K-theory is related to [[motivic cohomology]] (i.e. [[higher Chow groups]]) in degree n and weight n H{\tt \symbol{94}}n(X,Z(n)): for X=Spec(E), H{\tt \symbol{94}}n(X,Z(n)) is the nth Milnor K-group. This is how [[homotopy theory of schemes]] enters in the picture (one of the main feature introduced by Voevodsky to study motivic cohomology with finite coefficients is the theory of [[motivic Steenrod operations]]). On the other hand, Rost studied Milnor K-theory for itself: among a lot of other things, he proved that, if you consider it as a functor from the category of fields, with all its extra structures (residue maps interacting well), you can reconstruct [[higher Chow groups]] of schemes (over a field), via some Gersten complex.

Milnor K-theory is also a crucial ingredient in Kato's [[higher class field theory]].

\hypertarget{see_also}{}\subsection*{{See also}}\label{see_also}

\begin{itemize}%
\item [[Galois cohomology]]
\item [[motivic cohomology]]
\item [[Milnor conjecture]]
\item [[Bloch-Kato conjecture]]
\item [[motivic homotopy theory]]

\end{itemize}


\end{document}