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| {{for|Milnor's conjecture about the slice genus of torus knots|Milnor conjecture (topology)}}
| | Hello and welcome. My name is Irwin and I completely dig that title. Managing individuals is what I do and the salary has been truly satisfying. Years in the past we moved to North Dakota. To collect cash is one of the things I adore most.<br><br>my blog post - [http://www.pornextras.info/blog/187567 std testing at home] |
| In [[mathematics]], the '''Milnor conjecture''' was a proposal by {{harvs|txt|first=John|last= Milnor|year=1970|authorlink=John Milnor}} of a description of the [[Milnor K-theory]] (mod 2) of a general [[field (mathematics)|field]] ''F'' with [[characteristic (algebra)|characteristic]] different from 2, by means of the [[Galois cohomology|Galois]] (or equivalently [[étale cohomology|étale]]) cohomology of ''F'' with coefficients in '''Z'''/2'''Z'''. It was proved by {{harvs|txt|authorlink=Vladimir Voevodsky|first=Vladimir|last= Voevodsky|year1=1996|year2=2003a|year3=2003b}}.
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| ==Statement of the theorem==
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| Let ''F'' be a field of characteristic different from 2. Then there is an isomorphism
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| :<math>K_n^M(F)/2 \cong H_{\acute{e}t}^n(F, \mathbb{Z}/2\mathbb{Z})</math>
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| for all ''n'' ≥ 0, where ''K'' denotes the [[Milnor ring]].
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| ==About the proof==
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| The proof of this theorem by [[Vladimir Voevodsky]] uses several ideas developed by Voevodsky, [[Alexander Merkurjev]], [[Andrei Suslin]], [[Markus Rost]], [[Fabien Morel]], [[Eric Friedlander]], and others, including the newly minted theory of [[motivic cohomology]] (a kind of substitute for [[singular cohomology]] for [[algebraic varieties]]) and the [[motivic Steenrod algebra]].
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| ==Generalizations==
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| The analogue of this result for [[prime number|primes]] other than 2 was known as the [[Bloch–Kato conjecture]]. Work of Voevodsky and [[Markus Rost]] yielded a complete proof of this conjecture in 2009; the result is now called the [[norm residue isomorphism theorem]].
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| ==References==
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| * {{Citation | last1=Mazza | first1=Carlo | last2=Voevodsky | first2=Vladimir | author2-link=Vladimir Voevodsky | last3=Weibel | first3=Charles | title=Lecture notes on motivic cohomology | url=http://math.rutgers.edu/~weibel/motiviclectures.html | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Clay Mathematics Monographs | isbn=978-0-8218-3847-1 | mr=2242284 | year=2006 | volume=2 | author-link=Charles Weibel}}
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| * {{Citation | last1=Milnor | first1=John Willard | author1-link=John Milnor | title=Algebraic K-theory and quadratic forms | doi=10.1007/BF01425486 | mr=0260844 | year=1970 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=9 | pages=318–344 | issue=4}}
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| * {{citation | last1=Voevodsky | first1=Vladimir | author1-link=Vladimir Voevodsky | url=http://www.math.uiuc.edu/K-theory/0170 | title=The Milnor Conjecture | year=1996 | series=Preprint}}
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| * {{Citation | last1=Voevodsky | first1=Vladimir | author1-link=Vladimir Voevodsky | title=Reduced power operations in motivic cohomology | url=http://www.numdam.org/item?id=PMIHES_2003__98__1_0 | doi=10.1007/s10240-003-0009-z | mr=2031198 | year=2003a| journal=Institut des Hautes Études Scientifiques. Publications Mathématiques | issn=0073-8301 | issue=98 | pages=1–57 | volume=98}}
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| * {{Citation | last1=Voevodsky | first1=Vladimir | author1-link=Vladimir Voevodsky | title=Motivic cohomology with Z/2-coefficients | url=http://www.numdam.org/item?id=PMIHES_2003__98__59_0 | doi=10.1007/s10240-003-0010-6 | mr=2031199 | year=2003b | journal=Institut des Hautes Études Scientifiques. Publications Mathématiques | issn=0073-8301 | issue=98 | pages=59–104 | volume=98}}
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| ==Further reading==
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| * {{citation | last=Kahn | first=Bruno | chapter=La conjecture de Milnor (d'après V. Voevodsky) | language=French | editor1-last=Friedlander | editor1-first=Eric M. | editor2-last=Grayson | editor2-first=D.R. | title=Handbook of ''K''-theory | volume=2 | pages=1105–1149 | publisher=[[Springer-Verlag]] | year=2005 | isbn=3-540-23019-X | zbl=1101.19001 }}
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| [[Category:K-theory]]
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| [[Category:Conjectures]]
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| [[Category:Theorems in abstract algebra]]
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Hello and welcome. My name is Irwin and I completely dig that title. Managing individuals is what I do and the salary has been truly satisfying. Years in the past we moved to North Dakota. To collect cash is one of the things I adore most.
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