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In the mathematical field of [[algebraic K-theory]], the '''norm residue isomorphism theorem''' is a long-sought result whose  proof was completed in 2011. Its statement was previously known as the '''Bloch–Kato conjecture''', after [[Spencer Bloch]] and [[Kazuya Kato]], or more precisely the ''motivic'' Bloch–Kato conjecture in some places, since there is another Bloch–Kato conjecture on [[values of L-functions]].  The title "norm residue" originally referred to the symbol <math>(a_1,a_2)</math>
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taking values in the [[Brauer group]] of ''k'' (when the field contains all ℓ-th roots of unity). Its usage here is
in analogy with standard [[local class field theory]] which identifies the result in terms of a "higher" class field theory, still being developed.
 
It is a generalisation of the [[Milnor conjecture]] of K-theory, which was proved in the 1990s by [[Vladimir Voevodsky]], the Milnor conjecture being the 2-primary part of the Bloch–Kato conjecture. The point of the conjecture is to equate the torsion in a K-group of a [[field (mathematics)|field]] ''F'', algebraic information that is in general relatively inaccessible, with the torsion in a [[Galois cohomology]] group for ''F'', which is in many cases much easier to compute.  The '''Merkurjev–Suslin theorem''' is an intermediate result.  Now that the complete proof of the Bloch–Kato conjecture has been announced, due to several mathematicians and contained in quite a number of papers, the result is also known as the '''Voevodsky–Rost theorem''', for Voevodsky and [[Markus Rost]].
 
The norm residue isomorphism theorem implies the [[Quillen–Lichtenbaum conjecture]].
It is equivalent to a theorem whose statement was once referred to as the [[Norm_residue_isomorphism_theorem#Beilinson–Lichtenbaum conjecture|Beilinson–Lichtenbaum conjecture]].  
==Statement==
For any integer ℓ invertible in a field ''k'' there is a map
<math>\partial : k^*\rightarrow H^1(k,\mu_\ell) </math>
where <math>\mu_\ell</math> denotes the group of ℓ-th roots of unity in some separable extension of ''k''. 
It induces an isomorphism <math>k^*/k^{*\ell} \cong H^1(k,\mu_\ell)</math>. By multiplicativity, the map
&partial; extends to maps
 
:<math>\partial^n : k^* \times \cdots \times k^* \rightarrow H^n\left({k,\mu_\ell^{\otimes n}}\right) \  </math>
 
with the property that <math>\partial^n(\ldots,a,\ldots,1-a,\ldots)</math> vanishes.  This is the defining relation of Milnor's definition of "higher" ''K''-groups as the graded parts of the ring
 
:<math> K^M_*(k) := T^*(k^\times)/(a\otimes (1-a)) </math>,
 
the [[tensor algebra]] of the [[multiplicative group]] ''k''<sup>×</sup> by the [[two-sided ideal]], generated by the
 
:<math>\left \{a\otimes(1-a): \ a \neq 0,1 \right \}.</math>
 
Hence <math>\partial^n</math> may be regarded as a map on <math>K^M_n(k)</math>, called the ''Galois symbol'' or ''norm residue'' map.<ref name=Sri146>Srinivas (1996) p.146</ref><ref name=GS108>Gille & Szamuely (2006) p.108</ref><ref name=Efr221>Efrat (2006) p.221</ref>
 
The norm residue isomorphism theorem (or Bloch-Kato conjecture) states that for a field ''k'' and an integer ℓ that is invertible in ''k'', the norm residue map
:<math>\partial^n : K_n^M(k)/\ell \to H^n_{{\acute{\rm e}{\rm t}}}(k,\mu^{\otimes n}_\ell)</math>
from [[Milnor K-theory]]  mod l to [[étale cohomology]] is an isomorphism. The case ℓ=2 is the [[Milnor conjecture]] that was proved earlier by Voevodsky.  The case ''n''=2 is the Merkurjev–Suslin theorem.<ref name=Sri>Srinivas (1996) pp.145-193</ref><ref name=Efr221/>
 
==Beilinson-Lichtenbaum conjecture==
Let ''X'' be a smooth variety over a field containing <math>1/\ell</math>.
Beilinson and Lichtenbaum conjectured that the [[motivic cohomology]] group <math>H^{p,q}(X,Z/\ell)</math>
is isomorphic to the [[étale cohomology]] group <math>H^p_{{\acute{\rm e}{\rm t}}}(k,\mu^{\otimes q}_\ell)</math>
when ''p''&le;''q''.  This conjecture has now been proven, and is equivalent to the norm residue isomorphism theorem.
 
==References==
{{reflist}}
* {{cite book | last=Efrat | first=Ido | title=Valuations, orderings, and Milnor ''K''-theory | series=Mathematical Surveys and Monographs | volume=124 | location=Providence, RI | publisher=[[American Mathematical Society]] | year=2006 | isbn=0-8218-4041-X | zbl=1103.12002 }}
* {{cite book | last1=Gille | first1=Philippe | last2=Szamuely | first2=Tamás | title=Central simple algebras and Galois cohomology | series=Cambridge Studies in Advanced Mathematics | volume=101 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2006 | isbn=0-521-86103-9 | zbl=1137.12001 }}
* {{Cite web | last1=Rost | first1=Markus | title=Chain lemma for splitting fields of symbols | url=http://www.math.uni-bielefeld.de/~rost/chain-lemma.html | year=1998}}
* {{cite book | last=Srinivas | first=V. | title=Algebraic ''K''-theory | edition=Paperback reprint of the 1996 2nd | series=Modern Birkhäuser Classics | location=Boston, MA | publisher=[[Birkhäuser]] | year=2008 | isbn=978-0-8176-4736-0 | zbl=1125.19300 }}
* {{Cite arxiv | last1=Voevodsky | first1=Vladimir | author1-link=Vladimir Voevodsky | title=On motivic cohomology with Z/l coefficients | eprint=0805.4430 | year=2008 | class=math.AG}}
* {{Cite journal | last1=Weibel | first1=Charles | author1-link=Charles Weibel | title=The norm residue isomorphism theorem | doi=10.1112/jtopol/jtp013 | mr=2529300 | year=2009 | journal=[[Journal of Topology]] | volume=2 | issue=2 | pages=346–372}}
 
[[Category:Conjectures]]
[[Category:Algebraic K-theory]]
[[Category:Theorems in algebra]]

Latest revision as of 19:49, 7 January 2015

When it comes time to buy kitchen knives, there are numerous things that you need to have to take into consideration before merely purchasing the 1st set you come across. One particular of critical tools at the kitchen is knife. Along with a chef's knife, bigger units usually contain a bread knife, a longer serrated knife utilised to chop breads and unique meals with a difficult, outer shell and gentle inside. In addition they involve a carving knife, slicing knife and fairly a lot of unique meat knifes, which includes boning, fillet and cleaver knives. Specialised cutlery incorporates cheese and tomato knives, adorning knife and mincing knife. Once again, the knife should really really feel very good in your hand.

With our factory-direct goods, there's no require for you to worry about the good quality of the ceramic knife set totally free shipping reviews All you require to do is pick the 4pcs ceramic knife set critiques you want, and the service is assured. One particularthe highest rated knife sets that we can speak about right here is the J.A. Henckels International Forged Premio 18 Piece Block Set. The high quality of the knife you use is also vital.

And the handles are classic-style triple riveted which offers you full manage of every slice and movement you make with the Wusthof steak knives, never ever worry about dropping your knife once again or it wobbling about on you. On best of that they are created by Wusthof, which pretty much assure that you will be left with total control of the knife. Selecting the knife for the application is crucial as well.

That becoming stated, if you want a a lot more high-finish knife, you could do a complete lot worse than the $130 MAC MTH-80 The Victorinox is a great deal and a capable tool, but it performs at somewhere involving 80 and 90 % efficiency, compared to a high-efficiency knife like the MAC, which will bump that figure up to about 95 %. Maybe you are going to understand that you essentially favor a heavier knife.

This new edition of the well-liked original Hibben throwing knife options a higher tensile nylon cord wrapped handle. This substantial throwing Knife is made of all stainless steel building and comes with a sheath. This massive knife is wonderful for practicing your throwing technique. The Boker Magnum Bailey Ziell II Throwing Knife Set consists of three 13.25 inch knives. Each knife is constructed of super challenging .25 inch 420 J2 stainless steel. Every set incorporates a heavy duty nylon sheath.

In addition to relative ranks, a rating is assigned to every knife: U - Unacceptable S - Serviceable E - Exceptional - Outstanding. More force is necessary when making use of a serviceable knife than an great or outstanding one. The knives are properly balanced and sharp adequate to feel as if the user is merely guiding the knife and the knife is performing the cutting. I recommend going to a retailer where you can in fact hold the knife.

A chef's knife is high-priced but owning one particular superior chef's knife comes in handy for just about every cook. If cooking is one thing that you have become passionate about, then you might want to look at acquiring a chef knife bag. The excellent chef knife bag would be one particular that is each functional and versatile. When you loved this post and you would want to receive more info relating to Kitchen Knife review 2012 assure visit the page. The kitchen knife consists of three major functional parts the knife tip, the knife edge and the handle.

It simply suggests that as soon as an order comes, you set the pan and push it in the backburner until all the things of that order are ready to be Kitchen Devil Knife Set Review fired. US-MTA Adaptive knife techniques and applications are far more conducive to achievement in modern day aggressive scenarios. The mind set and programming of US-MTA Adaptive private protection is survival driven. Knife handles come in a assortment of materials like metal, plastic, resin, and wood. Really feel the balance of the knife in your hand.

Wusthof Classic Knives Set has the correct knife for all kitchen tasks from chopping onions and slicing cheeses and meats to cutting fruits and vegetables. Like other Wüsthof knives, and Classic Knives Set are produced of forged steel. Each and every knife blade is precision forged from a single piece of an exclusive high-carbon stainless steel with polypropylene handles triple riveted to the metal. There's absolutely nothing better than a set of Warther knives.

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