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Wikipedia says:

this circle of ideas is distinct from the Bloch–Kato conjecture of K-theory, extending the Milnor conjecture, a proof of which was announced in 2009

What exactly is the K-theory conjecture of Bloch-Kato and has it been proven?

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3 Answers

up vote 11 down vote accepted

Here are some lectures by Charles Weibel. Early on, they discuss Milnor Conjecture and Bloch-Kato, and they should go through the proof. My understanding is that there were a bunch of people involved in the proof, though a few were a bit reticent to actually write up their parts of it, and so Weibel drew the short straw and is the one writing it up.

EDIT: Adding a bit, here's Weibel's 2006 page where he notes the status as of then, and to make sure that this is roughly self-contained, here's the statement:

For an odd prime $\ell$, and a field $k$ containing $1/\ell$, the Milnor K-theory $K^M_n(k)/\ell$ is isomorphic to the étale cohomology $H^n_{ét}(k,μ_\ell^n)$ of the field $k$ with coefficients in the twists of $μ_\ell$.

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I think you mean Charles Weibel, not Chuck Weibel. –  Jonas Morrissey Jan 1 '10 at 21:48
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I've changed it, but he often goes by Chuck, and many people refer to him that way. –  Charles Siegel Jan 1 '10 at 21:52
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On 22 June 2013, Joël Riou is going to give a Bourbaki talk on

La conjecture de Bloch-Kato [d'après M. Rost et V. Voevodsky].

La conjecture de Bloch-Kato énonce que pour tout corps $k$ et tout nombre premier $l$ différent de la caractéristique de $k$, l'algèbre de K-théorie de Milnor de $k$ modulo $l$ (qui est définie par générateurs et relations) s'identifie à une algèbre de cohomologie galoisienne associée à $k$. La démonstration de cet énoncé, qui admet de nombreuses applications, utilise de façon essentielle d'une part les théories motiviques (cohomologie, homotopie, opérations de Steenrod) et d'autre part des constructions géometriques de variétés algébriques ayant des propriétés remarquables par rapport à des symboles en K-théorie de Milnor.

Usually the notes are put up on http://www.bourbaki.ens.fr/ some time after the actual talk. You can also write to Joël Riou directly at http://www.math.u-psud.fr/~riou/.

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Voevodsky gave a talk at a memorial conference for Grothendieck in January 2009 where he announced a full proof of the Bloch Kato Conjecture. The talk is available on google video.

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memorial conference for Grothendieck? what??? –  VA. Feb 9 '10 at 1:43
    
do you have a link? I couldn't find it... –  Sam Lichtenstein Feb 9 '10 at 2:39
    
Google Video tells me "Your search - voevodsky -"greg voevodsky" - did not match any videos.". This is because it's not spelled right. Look for "Voevodski": dailymotion.com/video/… –  Konrad Voelkel Feb 9 '10 at 9:21
    
Voevodsky spells his name with a "y" in his own papers so I'll go with that, but either way, to answer your question, type "IHES Grothendieck" into Google Video. You'll see many of the lectures pop up (Nori, Deligne, Voevodsky/i, etc). Or just follow Konrad V's link. –  Julian Feb 9 '10 at 23:09
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