# Milnor K-theory

In mathematics, Milnor K-theory is an invariant of fields defined by Milnor (1970). Originally viewed as an approximation to algebraic K-theory, Milnor K-theory has turned out to be an important invariant in its own right.

## Definition

The calculation of K2 of a field by 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 F× by the two-sided ideal generated by the elements

${\displaystyle a\otimes (1-a)\,}$

for a ≠ 0, 1 in F. The nth Milnor K-group KnM(F) is the nth graded piece of this graded ring; for example, K0M(F) = Z and K1M(F) = F*. There is a natural homomorphism

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

from the Milnor K-groups of a field to the Quillen K-groups, which is an isomorphism for n ≤ 2 but not for larger n, in general. For nonzero elements a1, ..., an in F, the symbol {a1, ..., an} in KnM(F) means the image of a1 ⊗ ... ⊗ an in the tensor algebra. Every element of Milnor K-theory can be written as a finite sum of symbols. The fact that {a, 1−a} = 0 in K2M(F) for a in F − {0,1} is sometimes called the Steinberg relation.

## Examples

We have ${\displaystyle K_{n}^{M}(\mathbb {F} _{q})=0}$ for n ≧ 2, while ${\displaystyle K_{2}^{M}(\mathbb {C} )}$ is an uncountable uniquely divisible group. (An abelian group is uniquely divisible if it is a vector space over the rational numbers.) 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 ${\displaystyle p}$.

## Applications

Milnor K-theory plays a fundamental role in higher class field theory, replacing K1M(F) = F* in the one-dimensional class field theory.

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

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

of the Milnor K-theory of a field with a certain motivic cohomology group.[2] 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 etale cohomology:

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

for any positive integer r invertible in the field F. This was proved by Voevodsky, with contributions by Rost and others.[3] This includes the MerkurjevSuslin theorem and the Milnor conjecture as special cases (the cases n = 2 and 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 W(F) → Z/2 given by the dimension of a quadratic form, modulo 2. Milnor defined a homomorphism from the mod 2 Milnor K-group KnM(F)/2 to the quotient In/In+1, sending a symbol {a1, ..., an} to the class of the n-fold Pfister form[4]

${\displaystyle \langle \langle a_{1},a_{2},...,a_{n}\rangle \rangle =\langle 1,-a_{1}\rangle \otimes \langle 1,-a_{2}\rangle \otimes ...\otimes \langle 1,-a_{n}\rangle .}$

Orlov, Vishik, and Voevodsky proved another statement called the Milnor conjecture, namely that this homomorphism KnM(F)/2 → In/In+1 is an isomorphism.[5]

## References

1. ^ Gille & Szamuely (2006), p. 184.
2. ^ Mazza, Voevodsky, Weibel (2005), Theorem 5.1.
3. ^ Voevodsky (2011).
4. ^ Elman, Karpenko, Merkurjev (2008), sections 5 and 9.B.
5. ^ Orlov, Vishik, Voevodsky (2007).