In mathematics, Milnor K-theory^[1] is an algebraic invariant (denoted ${\displaystyle K_{*}(F)}$ for a field ${\displaystyle F}$) defined by John Milnor (1970) as an attempt to study higher algebraic K-theory in the special case of fields. It was hoped this would help illuminate the structure for algebraic K-theory and give some insight about its relationships with other parts of mathematics, such as Galois cohomology and the Grothendieck–Witt ring of quadratic forms. Before Milnor K-theory was defined, there existed ad-hoc definitions for ${\displaystyle K_{1))$ and ${\displaystyle K_{2))$. Fortunately, it can be shown Milnor K-theory is a part of algebraic K-theory, which in general is the easiest part to compute.^[2]

Definition

Motivation

After the definition of the Grothendieck group ${\displaystyle K(R)}$ of a commutative ring, it was expected there should be an infinite set of invariants ${\displaystyle K_{i}(R)}$ called higher K-theory groups, from the fact there exists a short exact sequence

${\displaystyle K(R,I)\to K(R)\to K(R/I)\to 0}$

which should have a continuation by a long exact sequence. Note the group on the left is relative K-theory. This led to much study and as a first guess for what this theory would look like, Milnor gave a definition for fields. His definition is based upon two calculations of what higher K-theory "should" look like in degrees ${\displaystyle 1}$ and ${\displaystyle 2}$. Then, if in a later generalization of algebraic K-theory was given, if the generators of ${\displaystyle K_{*}(R)}$ lived in degree ${\displaystyle 1}$ and the relations in degree ${\displaystyle 2}$, then the constructions in degrees ${\displaystyle 1}$ and ${\displaystyle 2}$ would give the structure for the rest of the K-theory ring. Under this assumption, Milnor gave his "ad-hoc" definition. It turns out algebraic K-theory ${\displaystyle K_{*}(R)}$ in general has a more complex structure, but for fields the Milnor K-theory groups are contained in the general algebraic K-theory groups after tensoring with ${\displaystyle \mathbb {Q} }$, i.e. ${\displaystyle K_{n}^{M}(F)\otimes \mathbb {Q} \subseteq K_{n}(F)\otimes \mathbb {Q} }$.^[3] It turns out the natural map ${\displaystyle \lambda :K_{4}^{M}(F)\to K_{4}(F)}$ fails to be injective for a global field ${\displaystyle F}$^{[3]pg 96}.

Definition

Note for fields the Grothendieck group can be readily computed as ${\displaystyle K_{0}(F)=\mathbb {Z} }$ since the only finitely generated modules are finite-dimensional vector spaces. Also, Milnor's definition of higher K-groups depends upon the canonical isomorphism

${\displaystyle l\colon K_{1}(F)\to F^{*))$

(the group of units of ${\displaystyle F}$) and observing the calculation of K₂ of a field by Hideya Matsumoto, which gave the simple presentation

${\displaystyle K_{2}(F)={\frac {F^{*}\otimes F^{*)){\{l(a)\otimes l(1-a):a\neq 0,1\))))$

for a two-sided ideal generated by elements ${\displaystyle l(a)\otimes l(a-1)}$, called Steinberg relations. Milnor took the hypothesis that these were the only relations, hence he gave the following "ad-hoc" definition of Milnor K-theory as

${\displaystyle K_{n}^{M}(F)={\frac {K_{1}(F)\otimes \cdots \otimes K_{1}(F)}{\{l(a_{1})\otimes \cdots \otimes l(a_{n}):a_{i}+a_{i+1}=1\))}.}$

The direct sum of these groups is isomorphic to a tensor algebra over the integers of the multiplicative group ${\displaystyle K_{1}(F)\cong F^{*))$ modded out by the two-sided ideal generated by:

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

so

${\displaystyle \bigoplus _{n=0}^{\infty }K_{n}^{M}(F)\cong {\frac {T^{*}(K_{1}^{M}(F))}{\{l(a)\otimes l(1-a):a\neq 0,1\))))$

showing his definition is a direct extension of the Steinberg relations.

Properties

Ring structure

The graded module ${\displaystyle K_{*}^{M}(F)}$ is a graded-commutative ring^{[1]pg 1-3}.^[4] If we write

${\displaystyle (l(a_{1})\otimes \cdots \otimes l(a_{n}))\cdot (l(b_{1})\otimes \cdots \otimes l(b_{m}))}$

as

${\displaystyle l(a_{1})\otimes \cdots \otimes l(a_{n})\otimes l(b_{1})\otimes \cdots \otimes l(b_{m})}$

then for ${\displaystyle \xi \in K_{i}^{M}(F)}$ and ${\displaystyle \eta \in K_{j}^{M}(F)}$ we have

${\displaystyle \xi \cdot \eta =(-1)^{i\cdot j}\eta \cdot \xi .}$

From the proof of this property, there are some additional properties which fall out, like

$${\displaystyle l(a)^{2}=l(a)l(-1)}$$

${\displaystyle l(a)\in K_{1}(F)}$

${\displaystyle l(a)l(-a)=0}$

${\displaystyle a_{1}+\cdots +a_{n))$

${\displaystyle 0,1}$

$${\displaystyle l(a_{1})\cdots l(a_{n})=0}$$

${\displaystyle -1\in F}$

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

${\displaystyle \mathbb {Q} (i)}$

${\displaystyle \mathbb {Q} _{p}(i)}$

${\displaystyle {\sqrt {-1))\not \in \mathbb {Q} _{p))$

${\displaystyle F}$

Relation to Higher Chow groups and Quillen's higher K-theory

One of the core properties relating Milnor K-theory to higher algebraic K-theory is the fact there exists natural isomorphisms

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

$${\displaystyle K_{*}^{M}(F)\to {\text{CH))^{*}(F,*)}$$

$${\displaystyle \phi :F^{*}\to {\text{CH))^{1}(F,1)}$$

$${\displaystyle \phi (a)\phi (1-a)=0~{\text{in))~{\text{CH))^{2}(F,2)~{\text{for))~a,1-a\in F^{*))$$

$${\displaystyle {\begin{aligned}\{1\}&\mapsto 0\in {\text{CH))^{1}(F,1)\\\{a\}&\mapsto [a]\in {\text{CH))^{1}(F,1)\end{aligned))}$$

${\displaystyle [a]}$

${\displaystyle [a:1]\in \mathbb {P} _{F}^{1}-\{0,1,\infty \))$

${\displaystyle a\in F^{*}-\{1\))$

${\displaystyle [a]+[1/a]=0}$

${\displaystyle a\in F^{*}-\{1\))$

${\displaystyle [a]+[b]=[ab]}$

${\displaystyle [a]\cdot [b]}$

${\displaystyle {\text{CH))^{2}(F,2)}$

${\displaystyle b=1/a}$

${\displaystyle C^{1}(F,2)}$

${\displaystyle \partial }$

${\displaystyle [a]+[b]-[ab]}$

${\displaystyle ab\neq 1}$

${\displaystyle ab=1}$

${\displaystyle [a]-[1/a]}$

$${\displaystyle {\text{CH))^{p}(F,q)\otimes {\text{CH))^{r}(F,s)\to {\text{CH))^{p+r}(F,q+s)}$$

$${\displaystyle K_{*}^{M}(F)\to {\text{CH))^{*}(F,*)}$$

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

$${\displaystyle K_{n}(X)\otimes \mathbb {Q} \cong \bigoplus _{p}{\text{CH))^{p}(X,n)\otimes \mathbb {Q} }$$

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

from the Milnor K-groups of a field to the 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.

Representation in motivic cohomology

In motivic cohomology, specifically motivic homotopy theory, there is a sheaf ${\displaystyle K_{n,A))$ representing a generalization of Milnor K-theory with coefficients in an abelian group ${\displaystyle A}$. If we denote ${\displaystyle A_{tr}(X)=\mathbb {Z} _{tr}(X)\otimes A}$ then we define the sheaf ${\displaystyle K_{n,A))$ as the sheafification of the following pre-sheaf^{[5]pg 4}

$${\displaystyle K_{n,A}^{pre}:U\mapsto A_{tr}(\mathbb {A} ^{n})(U)/A_{tr}(\mathbb {A} ^{n}-\{0\})(U)}$$

${\displaystyle U\times \mathbb {A} ^{n))$

${\displaystyle A}$

${\displaystyle U}$

${\displaystyle \mathbb {Z} _{tr}(X)}$

${\displaystyle \mathbb {A} ^{1))$

${\displaystyle K(A,2n,n)}$

Examples

Finite fields

For a finite field ${\displaystyle F=\mathbb {F} _{q))$, ${\displaystyle K_{1}^{M}(F)}$ is a cyclic group of order ${\displaystyle q-1}$ (since is it isomorphic to ${\displaystyle \mathbb {F} _{q}^{*))$), so graded commutativity gives

$${\displaystyle l(a)\cdot l(b)=-l(b)\cdot l(a)}$$

$${\displaystyle l(a)^{2}=-l(a)^{2))$$

${\displaystyle K_{2}^{M}(F)}$

${\displaystyle \leq 2}$

${\displaystyle 1}$

${\displaystyle a,b\in F}$

${\displaystyle [a],[b]\in F/F^{\times 2))$

${\displaystyle 0}$

${\displaystyle a+b=1}$

${\displaystyle K_{2}^{M}(F)=0}$

${\displaystyle K_{n}^{M}(F)=0}$

${\displaystyle n>2}$

Real numbers

For the field of real numbers ${\displaystyle \mathbb {R} }$ the Milnor K-theory groups can be readily computed. In degree ${\displaystyle n}$ the group is generated by

$${\displaystyle K_{n}^{M}(\mathbb {R} )=\{(-1)^{n},l(a_{1})\cdots l(a_{n}):a_{1},\ldots ,a_{n}>0\))$$

${\displaystyle (-1)^{n))$

${\displaystyle 2}$

${\displaystyle l(a_{1})\cdots l(a_{n})}$

${\displaystyle (-1)^{n))$

Other calculations

${\displaystyle K_{2}^{M}(\mathbb {C} )}$ is an uncountable uniquely divisible group.^[7] 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 ${\displaystyle p}$. For ${\displaystyle n\geq 3}$, ${\displaystyle K_{n}^{M}(\mathbb {Q} )\cong \mathbb {Z} /2}$. The full proof is in the appendix of Milnor's original paper.^[1] Some of the computation can be seen by looking at a map on ${\displaystyle K_{2}^{M}(F)}$ induced from the inclusion of a global field ${\displaystyle F}$ to its completions ${\displaystyle F_{v))$, so there is a morphism

$${\displaystyle K_{2}^{M}(F)\to \bigoplus _{v}K_{2}^{M}(F_{v})/({\text{max. divis. subgr.)))}$$

${\displaystyle F}$

In addition, for a general local field ${\displaystyle F}$ (such as a finite extension ${\displaystyle K/\mathbb {Q} _{p))$), the Milnor K-groups ${\displaystyle K_{n}^{M}(F)}$ are divisible.

K_*^M(F(t))

There is a general structure theorem computing ${\displaystyle K_{n}^{M}(F(t))}$ for a field ${\displaystyle F}$ in relation to the Milnor K-theory of ${\displaystyle F}$ and extensions ${\displaystyle F[t]/(\pi )}$ for non-zero primes ideals ${\displaystyle (\pi )\in {\text{Spec))(F[t])}$. This is given by an exact sequence

$${\displaystyle 0\to K_{n}^{M}(F)\to K_{n}^{M}(F(t))\xrightarrow {\partial _{\pi )) \bigoplus _{(\pi )\in {\text{Spec))(F[t])}K_{n-1}F[t]/(\pi )\to 0}$$

${\displaystyle \partial _{\pi }:K_{n}^{M}(F(t))\to K_{n-1}F[t]/(\pi )}$

${\displaystyle F}$

${\displaystyle {\overline {F))_{v))$

${\displaystyle v}$

$${\displaystyle \partial :K_{n}^{M}(F)\to K_{n-1}^{M}({\overline {F)))}$$

${\displaystyle U\subset F}$

${\displaystyle 0}$

$${\displaystyle U\to {\overline {F))_{v}^{*))$$

${\displaystyle u\mapsto {\overline {u))}$

$${\displaystyle \partial (l(\pi )l(u_{2})\cdots l(u_{n}))=l({\overline {u))_{2})\cdots l({\overline {u))_{n})}$$

${\displaystyle \pi }$

${\displaystyle {\text{Ord))_{v}(\pi )=1}$

$${\displaystyle \partial (l(u_{1})\cdots l(u_{n}))=0}$$

${\displaystyle (\pi )\in {\text{Spec))(F[t])}$

${\displaystyle v_{\pi }:F(t)\to F[t]/(\pi )}$

${\displaystyle \partial _{\pi ))$

Applications

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.^[8] 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.^[9] 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.^[10]

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.^[11]