Milnor K-theory

The calculation of K₂ of a field by Hideya Matsumoto led Milnor to the following, seemingly naive, definition of the "higher" K-groups of a field F:

${\displaystyle K_{*}^{M}(F):=T^{*}F^{\times }/(a\otimes (1-a)),}$

the quotient of the tensor algebra over the integers of the multiplicative group ${\displaystyle F^{\times }}$ by the two-sided ideal generated by:

${\displaystyle \left\{a\otimes (1-a):0,1\neq a\in F\right\}.}$

The nth Milnor K-group ${\displaystyle K_{n}^{M}(F)}$ is the nth graded piece of this graded ring; for example, ${\displaystyle K_{0}^{M}(F)=\mathbb {Z} }$ and ${\displaystyle K_{1}^{M}(F)=F^{\times }.}$ There is a natural homomorphism

${\displaystyle K_{n}^{M}(F)\to K_{n}(F)}$

from the Milnor K-groups of a field to the Daniel Quillen K-groups, which is an isomorphism for ${\displaystyle n\leq 2}$ but not for larger n, in general. For nonzero elements ${\displaystyle a_{1},\ldots ,a_{n}}$ in F, the symbol ${\displaystyle \{a_{1},\ldots ,a_{n}\}}$ in ${\displaystyle K_{n}^{M}(F)}$ means the image of ${\displaystyle a_{1}\otimes \cdots \otimes a_{n}}$ in the tensor algebra. Every element of Milnor K-theory can be written as a finite sum of symbols. The fact that ${\displaystyle \{a,1-a\}=0}$ in ${\displaystyle K_{2}^{M}(F)}$ for ${\displaystyle a\in F\setminus \{0,1\}}$ is sometimes called the Steinberg relation.

The ring ${\displaystyle K_{*}^{M}(F)}$ is graded-commutative.[1]

We have ${\displaystyle K_{n}^{M}(\mathbb {F} _{q})=0}$ for ${\displaystyle n>2}$, while ${\displaystyle K_{2}^{M}(\mathbb {C} )}$ is an uncountable uniquely divisible group.[2] Also, ${\displaystyle K_{2}^{M}(\mathbb {R} )}$ is the direct sum of a cyclic group of order 2 and an uncountable uniquely divisible group; ${\displaystyle K_{2}^{M}(\mathbb {Q} _{p})}$ is the direct sum of the multiplicative group of ${\displaystyle \mathbb {F} _{p}}$ and an uncountable uniquely divisible group; ${\displaystyle K_{2}^{M}(\mathbb {Q} )}$ is the direct sum of the cyclic group of order 2 and cyclic groups of order ${\displaystyle p-1}$ for all odd prime ${\displaystyle p}$.

Milnor K-theory plays a fundamental role in higher class field theory, replacing ${\displaystyle K_{1}^{M}(F)=F^{\times }}$ in the one-dimensional class field theory.

Milnor K-theory fits into the broader context of motivic cohomology, via the isomorphism

${\displaystyle K_{n}^{M}(F)\cong H^{n}(F,\mathbb {Z} (n))}$

of the Milnor K-theory of a field with a certain motivic cohomology group.[3] In this sense, the apparently ad hoc definition of Milnor K-theory becomes a theorem: certain motivic cohomology groups of a field can be explicitly computed by generators and relations.

A much deeper result, the Bloch-Kato conjecture (also called the norm residue isomorphism theorem), relates Milnor K-theory to Galois cohomology or étale cohomology:

${\displaystyle K_{n}^{M}(F)/r\cong H_{\mathrm {et} }^{n}(F,\mathbb {Z} /r(n)),}$

for any positive integer r invertible in the field F. This conjecture was proved by Vladimir Voevodsky, with contributions by Markus Rost and others.[4] This includes the theorem of Alexander Merkurjev and Andrei Suslin as well as the Milnor conjecture as special cases (the cases when ${\displaystyle n=2}$ and ${\displaystyle r=2}$, respectively).

Finally, there is a relation between Milnor K-theory and quadratic forms. For a field F of characteristic not 2, define the fundamental ideal I in the Witt ring of quadratic forms over F to be the kernel of the homomorphism ${\displaystyle W(F)\to \mathbb {Z} /2}$ given by the dimension of a quadratic form, modulo 2. Milnor defined a homomorphism:

${\displaystyle {\begin{cases}K_{n}^{M}(F)/2\to I^{n}/I^{n+1}\\\{a_{1},\ldots ,a_{n}\}\mapsto \langle \langle a_{1},\ldots ,a_{n}\rangle \rangle =\langle 1,-a_{1}\rangle \otimes \cdots \otimes \langle 1,-a_{n}\rangle \end{cases}}}$

where ${\displaystyle \langle \langle a_{1},a_{2},\ldots ,a_{n}\rangle \rangle }$ denotes the class of the n-fold Pfister form.[5]

Dmitri Orlov, Alexander Vishik, and Voevodsky proved another statement called the Milnor conjecture, namely that this homomorphism ${\displaystyle K_{n}^{M}(F)/2\to I^{n}/I^{n+1}}$ is an isomorphism.[6]

• Azumaya algebra

• Elman, Richard; Karpenko, Nikita; Merkurjev, Alexander (2008), Algebraic and geometric theory of quadratic forms, American Mathematical Society, ISBN 978-0-8218-4329-1, MR 2427530
• Gille, Philippe; Szamuely, Tamás (2006). Central simple algebras and Galois cohomology. Cambridge Studies in Advanced Mathematics. 101. Cambridge: Cambridge University Press. ISBN 0-521-86103-9. MR 2266528. Zbl 1137.12001.
• Mazza, Carlo; Voevodsky, Vladimir; Weibel, Charles (2006), Lectures in Motivic Cohomology, Clay Mathematical Monographs, 2, American Mathematical Society, ISBN 978-0-8218-3847-1, MR 2242284
• Milnor, John Willard (1970), With an appendix by John Tate, "Algebraic K-theory and quadratic forms", Inventiones Mathematicae, 9: 318–344, Bibcode:1970InMat...9..318M, doi:10.1007/BF01425486, ISSN 0020-9910, MR 0260844, Zbl 0199.55501
• Orlov, Dmitri; Vishik, Alexander; Voevodsky, Vladimir (2007), "An exact sequence for ${\displaystyle K_{*}^{M}/2}$ with applications to quadratic forms", Annals of Mathematics, 165: 1–13, arXiv:math/0101023, doi:10.4007/annals.2007.165.1, MR 2276765
• Voevodsky, Vladimir (2011), "On motivic cohomology with ${\displaystyle \mathbb {Z} /\ell }$-coefficients", Annals of Mathematics, 174 (1): 401–438, arXiv:0805.4430, doi:10.4007/annals.2011.174.1.11, MR 2811603