Milnor K-theory

In mathematics, Milnor K-theory is an algebraic invariant (denoted $K_{*}(F)$ for a field $F$) 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 $K_{1}$ and $K_{2}$. Fortunately, it can be shown Milnor K-theory is a part of algebraic K-theory, which in general is the easiest part to compute.

Definition

Motivation

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

$K(R,I)\to K(R)\to K(R/I)\to 0$

which should have a continuation by a long exact sequence. Note the group on the left is relative K-theory. This lead to much study and as a first guess for what this theory would look like, Milnor gave a definition for fields.

Definition

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

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

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

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

for a two-sided ideal generated by elements $l(a)\otimes l(a-1)$, 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

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

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

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

so

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

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

Properties

Ring structure

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

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

as

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

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

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

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

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

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

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

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

Relation to Quillen's higher K-theory

In addition, there is a natural homomorphism

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

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

Examples

Finite fields

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

$l(a)\cdot l(b)=-l(b)\cdot l(a)$

hence

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

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

Real numbers

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

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

where $(-1)^{n}$ gives a group of order $2$ and the subgroup generated by the $l(a_{1})\cdots l(a_{n})$ is divisible. The subgroup generated by $(-1)^{n}$ 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. The others are lifts from the classical Steenrod operations to motivic cohomology.

Other calculations

$K_{2}^{M}(\mathbb {C} )$ is an uncountable uniquely divisible group. Also, $K_{2}^{M}(\mathbb {R} )$ is the direct sum of a cyclic group of order 2 and an uncountable uniquely divisible group; $K_{2}^{M}(\mathbb {Q} _{p})$ is the direct sum of the multiplicative group of $\mathbb {F} _{p}$ and an uncountable uniquely divisible group; $K_{2}^{M}(\mathbb {Q} )$ is the direct sum of the cyclic group of order 2 and cyclic groups of order $p-1$ for all odd prime $p$. For $n\geq 3$, $K_{n}^{M}(\mathbb {Q} )\cong \mathbb {Z} /2$. The full proof is in the appendix of Milnor's original paper. Some of the computation can be seen by looking at a map on $K_{2}^{M}(F)$ induced from the inclusion of a global field $F$ to its completions $F_{v}$, so there is a morphism

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

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

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

K_*^M(F(t))

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

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

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

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

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

$U\to {\overline {F}}_{v}^{*}$ where $u\mapsto {\overline {u}}$

we have

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

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

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

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

Applications

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

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

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

of the Milnor K-theory of a field with a certain motivic cohomology group. 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:

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

for any positive integer r invertible in the field F. This conjecture was proved by Vladimir Voevodsky, with contributions by Markus Rost and others. This includes the theorem of Alexander Merkurjev and Andrei Suslin as well as the Milnor conjecture as special cases (the cases when $n=2$ 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 $W(F)\to \mathbb {Z} /2$ given by the dimension of a quadratic form, modulo 2. Milnor defined a homomorphism:

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

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

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

See also

• Azumaya algebra
• Motivic homotopy theory

References

• 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. Vol. 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, vol. 2, American Mathematical Society, ISBN 978-0-8218-3847-1, MR 2242284
• Milnor, John Willard (1970), "Algebraic K-theory and quadratic forms", Inventiones Mathematicae, 9, With an appendix by John Tate: 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 $K_{*}^{M}/2$ 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 $\mathbb {Z} /\ell$-coefficients", Annals of Mathematics, 174 (1): 401–438, arXiv:0805.4430, doi:10.4007/annals.2011.174.1.11, MR 2811603