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In mathematics, '''Milnor K-theory''' was an early attempt to define higher [[algebraic K-theory]], introduced by {{harvs|txt|last=Milnor|year=1970|authorlink=John Milnor}}.


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==Definition==
 
The calculation of ''K''<sub>2</sub> of a field ''F'' led Milnor to the following ad hoc definition of "higher" ''K''-groups by
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:<math> K^M_*(F) := T^*F^\times/(a\otimes (1-a)), \, </math>


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thus as graded parts of a quotient of the [[tensor algebra]] of the [[multiplicative group]] ''F''<sup>×</sup> by the [[two-sided ideal]], generated by the  
 
:<math>a\otimes(1-a) \, </math>
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for ''a''&nbsp;≠&nbsp;0,&nbsp;1. For ''n''&nbsp;=&nbsp;0,1,2 these coincide with Quillen's K-groups of a field, but for ''n''&nbsp;≧&nbsp;3 they differ in general. We define the '''symbol''' <math>\{a_1,\ldots,a_n\}</math> as the image of <math>a_1 \otimes \cdots \otimes a_n</math>: the case ''n''=2 is a [[Steinberg symbol]].<ref name=Lam366/>
 
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The tensor product on the tensor algebra induces a product <math> K_m \times K_n \rightarrow K_{m+n}</math> making <math> K^M_*(F)</math> a [[graded ring]] which is [[graded-commutative]].<ref name=GS184>Gille & Szamuely (2006) p.184</ref>


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==Examples==
 
For example, we have <math>K^M_n(\mathbb{F}_q) = 0</math> for&nbsp;''n''&nbsp;≧&nbsp;2;
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<math>K^M_2(\mathbb{C})</math> is an uncountable uniquely divisible group;
 
<math>K^M_2(\mathbb{R})</math> is the direct sum of a cyclic group of order 2 and an uncountable uniquely divisible group;
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<math>K^M_2(\mathbb{Q}_p)</math> is the direct sum of the multiplicative group of <math>\mathbb{F}_p</math> and an uncountable uniquely divisible group;
 
<math>K^M_2(\mathbb{Q})</math> is the direct sum of the cyclic group of order 2 and cyclic groups of order <math>p-1</math> for all odd prime <math>p</math>.
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==Applications==
Milnor K-theory plays a fundamental role in higher class field theory, replacing <math>K^M_1</math> in the one dimensional [[class field theory]].


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Milnor K-theory modulo 2, denoted ''k''<sub>*</sub>(''F'') is related to [[étale cohomology|étale]] (or [[Galois cohomology|Galois]]) cohomology of the field ''F'' by the [[Milnor conjecture]], proven by Voevodsky. The analogous statement for odd primes is the [[Bloch–Kato conjecture]], proved by Voevodsky, Rost, and others.
 
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There are homomorphisms from ''k''<sub>''n''</sub>(''F'') to the [[Witt ring (forms)|Witt ring]] of ''F'' by taking the symbol


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:<math> \{a_1,\ldots,a_n\} \mapsto \langle \langle a_1, a_2, ... , a_n \rangle \rangle
 
= \langle 1, a_1 \rangle \otimes \langle 1, a_2 \rangle \otimes ... \otimes \langle 1, a_n \rangle \ ,</math>
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where the image is a [[Pfister form]] of dimension 2<sup>''n''</sup>.<ref name=Lam366>Lam (2005) p.366</ref>  The image can be taken as ''I''<sup>''n''</sup>/''I''<sup>''n''+1</sup> and the map is surjective since the Pfister forms additively generate ''I''<sup>''n''</sup>.<ref name=Lam316>Lam (2005) p.316</ref> The Milnor conjecture can be interpreted as stating that these maps are isomorphisms.<ref name=Lam366/>


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==References==
 
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* {{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 book | title=Introduction to Quadratic Forms over Fields | volume=67 | series=Graduate Studies in Mathematics | first=Tsit-Yuen | last=Lam | authorlink=Tsit Yuen Lam | publisher=[[American Mathematical Society]] | year=2005 | isbn=0-8218-1095-2 | zbl=1068.11023 | mr = 2104929 }}
  <li>[http://www.ultimate.cs2d.net/index.php?site=gallery&picID=33 http://www.ultimate.cs2d.net/index.php?site=gallery&picID=33]</li>
* {{Citation | last1=Milnor | first1=John Willard | author1-link= John Milnor | title=Algebraic ''K''-theory and quadratic forms | others=With an appendix by J. Tate | mr=0260844 | zbl=0199.55501  | year=1970 | journal=[[Inventiones Mathematicae]] | issn=0020-9910 | volume=9 | pages=318–344 | doi=10.1007/BF01425486}}
 
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==Further reading==
* {{citation | 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 }}


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Latest revision as of 05:39, 24 September 2012

In mathematics, Milnor K-theory was an early attempt to define higher algebraic K-theory, introduced by Template:Harvs.

Definition

The calculation of K2 of a field F led Milnor to the following ad hoc definition of "higher" K-groups by

K*M(F):=T*F×/(a(1a)),

thus as graded parts of a quotient of the tensor algebra of the multiplicative group F× by the two-sided ideal, generated by the

a(1a)

for a ≠ 0, 1. For n = 0,1,2 these coincide with Quillen's K-groups of a field, but for n ≧ 3 they differ in general. We define the symbol {a1,,an} as the image of a1an: the case n=2 is a Steinberg symbol.[1]

The tensor product on the tensor algebra induces a product Km×KnKm+n making K*M(F) a graded ring which is graded-commutative.[2]

Examples

For example, we have KnM(𝔽q)=0 for n ≧ 2; K2M() is an uncountable uniquely divisible group; K2M() is the direct sum of a cyclic group of order 2 and an uncountable uniquely divisible group; K2M(p) is the direct sum of the multiplicative group of 𝔽p and an uncountable uniquely divisible group; K2M() is the direct sum of the cyclic group of order 2 and cyclic groups of order p1 for all odd prime p.

Applications

Milnor K-theory plays a fundamental role in higher class field theory, replacing K1M in the one dimensional class field theory.

Milnor K-theory modulo 2, denoted k*(F) is related to étale (or Galois) cohomology of the field F by the Milnor conjecture, proven by Voevodsky. The analogous statement for odd primes is the Bloch–Kato conjecture, proved by Voevodsky, Rost, and others.

There are homomorphisms from kn(F) to the Witt ring of F by taking the symbol

{a1,,an}a1,a2,...,an=1,a11,a2...1,an,

where the image is a Pfister form of dimension 2n.[1] The image can be taken as In/In+1 and the map is surjective since the Pfister forms additively generate In.[3] The Milnor conjecture can be interpreted as stating that these maps are isomorphisms.[1]

References

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  1. 1.0 1.1 1.2 Lam (2005) p.366
  2. Gille & Szamuely (2006) p.184
  3. Lam (2005) p.316