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.

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

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

for ${\displaystyle l(a)\in K_{1}(F)}$ since ${\displaystyle l(a)l(-a)=0}$. Also, if ${\displaystyle a_{1}+\cdots +a_{n}}$ of non-zero fields elements equals ${\displaystyle 0,1}$, then

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

There's a direct arithmetic application: ${\displaystyle -1\in F}$ is a sum of squares if and only if every positive dimensional ${\displaystyle K_{n}^{M}(F)}$ is nilpotent, which is a powerful statement about the structure of Milnor K-groups. In particular, for the fields ${\displaystyle \mathbb {Q} (i)}$, ${\displaystyle \mathbb {Q} _{p}(i)}$ with ${\displaystyle {\sqrt {-1}}\not \in \mathbb {Q} _{p}}$, all of its Milnor K-groups are nilpotent. In the converse case, the field ${\displaystyle F}$ can be embedded into a real closed field, which gives a total ordering on the field.

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

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

to Bloch's Higher chow groups which induces a morphism of graded rings

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

This can be verified using an explicit morphism^{[2]pg 181}

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

where

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

This map is given by

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

for ${\displaystyle [a]}$ the class of the point ${\displaystyle [a:1]\in \mathbb {P} _{F}^{1}-\{0,1,\infty \}}$ with ${\displaystyle a\in F^{*}-\{1\}}$. The main property to check is that ${\displaystyle [a]+[1/a]=0}$ for ${\displaystyle a\in F^{*}-\{1\}}$ and ${\displaystyle [a]+[b]=[ab]}$. Note this is distinct from ${\displaystyle [a]\cdot [b]}$ since this is an element in ${\displaystyle {\text{CH}}^{2}(F,2)}$. Also, the second property implies the first for ${\displaystyle b=1/a}$. This check can be done using a rational curve defining a cycle in ${\displaystyle C^{1}(F,2)}$ whose image under the boundary map ${\displaystyle \partial }$ is the sum ${\displaystyle [a]+[b]-[ab]}$for ${\displaystyle ab\neq 1}$, showing they differ by a boundary. Similarly, if ${\displaystyle ab=1}$ the boundary map sends this cycle to ${\displaystyle [a]-[1/a]}$, showing they differ by a boundary. The second main property to show is the Steinberg relations. With these, and the fact the higher Chow groups have a ring structure

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

we get an explicit map

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

Showing the map in the reverse direction is an isomorphism is more work, but we get the isomorphisms

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

We can then relate the higher Chow groups to higher algebraic K-theory using the fact there are isomorphisms

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

giving the relation to Quillen's higher algebraic K-theory. Note that the maps

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

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}

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

Note that sections of this pre-sheaf are equivalent classes of cycles on ${\displaystyle U\times \mathbb {A} ^{n}}$ with coefficients in ${\displaystyle A}$ which are equidimensional and finite over ${\displaystyle U}$ (which follows straight from the definition of ${\displaystyle \mathbb {Z} _{tr}(X)}$). It can be shown there is an ${\displaystyle \mathbb {A} ^{1}}$-weak equivalence with the motivic Eilenberg-Maclane sheaves ${\displaystyle K(A,2n,n)}$ (depending on the grading convention).

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

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

hence

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

Because ${\displaystyle K_{2}^{M}(F)}$ is a finite group, this implies it must have order ${\displaystyle \leq 2}$. Looking further, ${\displaystyle 1}$ can always be expressed as a sum of quadratic non-residues, i.e. elements ${\displaystyle a,b\in F}$ such that ${\displaystyle [a],[b]\in F/F^{\times 2}}$ are not equal to ${\displaystyle 0}$, hence ${\displaystyle a+b=1}$ showing ${\displaystyle K_{2}^{M}(F)=0}$. Because the Steinberg relations generate all relations in the Milnor K-theory ring, we have ${\displaystyle K_{n}^{M}(F)=0}$ for ${\displaystyle n>2}$.

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

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

where ${\displaystyle (-1)^{n}}$ gives a group of order ${\displaystyle 2}$ and the subgroup generated by the ${\displaystyle l(a_{1})\cdots l(a_{n})}$ is divisible. The subgroup generated by ${\displaystyle (-1)^{n}}$ is not divisible because otherwise it could be expressed as a sum of squares. The Milnor K-theory ring is important in the study of motivic homotopy theory because it gives generators for part of the motivic Steenrod algebra.^[6] The others are lifts from the classical Steenrod operations to motivic cohomology.

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

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

whose kernel finitely generated. In addition, the cokernel is isomorphic to the roots of unity in ${\displaystyle F}$.

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

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

where ${\displaystyle \partial _{\pi }:K_{n}^{M}(F(t))\to K_{n-1}F[t]/(\pi )}$ is a morphism constructed from a reduction of ${\displaystyle F}$ to ${\displaystyle {\overline {F}}_{v}}$ for a discrete valuation ${\displaystyle v}$. This follows from the theorem there exists only one homomorphism

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

which for the group of units ${\displaystyle U\subset F}$ which are elements have valuation ${\displaystyle 0}$, having a natural morphism

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

where ${\displaystyle u\mapsto {\overline {u}}}$ we have

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

where ${\displaystyle \pi }$ a prime element, meaning ${\displaystyle {\text{Ord}}_{v}(\pi )=1}$, and

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

Since every non-zero prime ideal ${\displaystyle (\pi )\in {\text{Spec}}(F[t])}$ gives a valuation ${\displaystyle v_{\pi }:F(t)\to F[t]/(\pi )}$, we get the map ${\displaystyle \partial _{\pi }}$ on the Milnor K-groups.