Milnor K-theory

In mathematics, Milnor K-theory^[1] is an algebraic invariant (denoted [math]\displaystyle{ K_*(F) }[/math] for a field [math]\displaystyle{ F }[/math]) 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 [math]\displaystyle{ K_1 }[/math] and [math]\displaystyle{ K_2 }[/math]. 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 [math]\displaystyle{ K(R) }[/math] of a commutative ring, it was expected there should be an infinite set of invariants [math]\displaystyle{ K_i(R) }[/math] called higher K-theory groups, from the fact there exists a short exact sequence

[math]\displaystyle{ K(R,I) \to K(R) \to K(R/I) \to 0 }[/math]

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 [math]\displaystyle{ 1 }[/math] and [math]\displaystyle{ 2 }[/math]. Then, if in a later generalization of algebraic K-theory was given, if the generators of [math]\displaystyle{ K_*(R) }[/math] lived in degree [math]\displaystyle{ 1 }[/math] and the relations in degree [math]\displaystyle{ 2 }[/math], then the constructions in degrees [math]\displaystyle{ 1 }[/math] and [math]\displaystyle{ 2 }[/math] 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 [math]\displaystyle{ K_*(R) }[/math] 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 [math]\displaystyle{ \mathbb{Q} }[/math], i.e. [math]\displaystyle{ K^M_n(F)\otimes \mathbb{Q} \subseteq K_n(F)\otimes \mathbb{Q} }[/math].^[3] It turns out the natural map [math]\displaystyle{ \lambda:K^M_4(F) \to K_4(F) }[/math] fails to be injective for a global field [math]\displaystyle{ F }[/math]^{[3]pg 96}.

Definition

Note for fields the Grothendieck group can be readily computed as [math]\displaystyle{ K_0(F) = \mathbb{Z} }[/math] since the only finitely generated modules are finite-dimensional vector spaces. Also, Milnor's definition of higher K-groups depends upon the canonical isomorphism

[math]\displaystyle{ l\colon K_1(F) \to F^* }[/math]

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

[math]\displaystyle{ K_2(F) = \frac{F^*\otimes F^*}{\{l(a)\otimes l(1-a) : a \neq 0 ,1\}} }[/math]

for a two-sided ideal generated by elements [math]\displaystyle{ l(a)\otimes l(a-1) }[/math], 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

[math]\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 \}}. }[/math]

The direct sum of these groups is isomorphic to a tensor algebra over the integers of the multiplicative group [math]\displaystyle{ K_1(F) \cong F^* }[/math] modded out by the two-sided ideal generated by:

[math]\displaystyle{ \left \{ l(a)\otimes l(1-a) : 0,1 \neq a \in F \right \} }[/math]

so

[math]\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\}} }[/math]

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

Properties

Ring structure

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

[math]\displaystyle{ (l(a_1)\otimes\cdots \otimes l(a_n))\cdot (l(b_1)\otimes\cdots \otimes l(b_m)) }[/math]

as

[math]\displaystyle{ l(a_1)\otimes\cdots \otimes l(a_n) \otimes l(b_1)\otimes\cdots \otimes l(b_m) }[/math]

then for [math]\displaystyle{ \xi \in K_i^M(F) }[/math] and [math]\displaystyle{ \eta \in K^M_j(F) }[/math] we have

[math]\displaystyle{ \xi \cdot \eta = (-1)^{i \cdot j}\eta \cdot \xi. }[/math]

From the proof of this property, there are some additional properties which fall out, like [math]\displaystyle{ l(a)^2 = l(a)l(-1) }[/math] for [math]\displaystyle{ l(a) \in K_1(F) }[/math] since [math]\displaystyle{ l(a)l(-a) = 0 }[/math]. Also, if [math]\displaystyle{ a_1+\cdots + a_n }[/math] of non-zero fields elements equals [math]\displaystyle{ 0,1 }[/math], then [math]\displaystyle{ l(a_1)\cdots l(a_n) = 0 }[/math] There's a direct arithmetic application: [math]\displaystyle{ -1 \in F }[/math] is a sum of squares if and only if every positive dimensional [math]\displaystyle{ K_n^M(F) }[/math] is nilpotent, which is a powerful statement about the structure of Milnor K-groups. In particular, for the fields [math]\displaystyle{ \mathbb{Q}(i) }[/math], [math]\displaystyle{ \mathbb{Q}_p(i) }[/math] with [math]\displaystyle{ \sqrt{-1} \not\in \mathbb{Q}_p }[/math], all of its Milnor K-groups are nilpotent. In the converse case, the field [math]\displaystyle{ F }[/math] 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 [math]\displaystyle{ K_n^M(F) \to \text{CH}^{n}(F,n) }[/math] to Bloch's Higher chow groups which induces a morphism of graded rings [math]\displaystyle{ K_*^M(F) \to \text{CH}^*(F,*) }[/math] This can be verified using an explicit morphism^{[2]pg 181} [math]\displaystyle{ \phi:F^* \to \text{CH}^1(F,1) }[/math] where [math]\displaystyle{ \phi(a)\phi(1-a) = 0 ~\text{in}~ \text{CH}^2(F,2) ~\text{for}~ a,1-a \in F^* }[/math] This map is given by [math]\displaystyle{ \begin{align} \{1\} &\mapsto 0 \in \text{CH}^1(F,1) \\ \{a\} &\mapsto [a] \in \text{CH}^1(F,1) \end{align} }[/math] for [math]\displaystyle{ [a] }[/math] the class of the point [math]\displaystyle{ [a:1] \in \mathbb{P}^1_F-\{0,1,\infty \} }[/math] with [math]\displaystyle{ a \in F^*-\{1\} }[/math]. The main property to check is that [math]\displaystyle{ [a] + [1/a] = 0 }[/math] for [math]\displaystyle{ a \in F^*-\{1\} }[/math] and [math]\displaystyle{ [a] + [b] = [ab] }[/math]. Note this is distinct from [math]\displaystyle{ [a]\cdot [b] }[/math] since this is an element in [math]\displaystyle{ \text{CH}^2(F,2) }[/math]. Also, the second property implies the first for [math]\displaystyle{ b = 1/a }[/math]. This check can be done using a rational curve defining a cycle in [math]\displaystyle{ C^1(F,2) }[/math] whose image under the boundary map [math]\displaystyle{ \partial }[/math] is the sum [math]\displaystyle{ [a] + [b] - [ab] }[/math]for [math]\displaystyle{ ab \neq 1 }[/math], showing they differ by a boundary. Similarly, if [math]\displaystyle{ ab=1 }[/math] the boundary map sends this cycle to [math]\displaystyle{ [a] - [1/a] }[/math], 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 [math]\displaystyle{ \text{CH}^p(F,q) \otimes \text{CH}^r(F,s) \to \text{CH}^{p+r}(F,q+s) }[/math] we get an explicit map [math]\displaystyle{ K_*^M(F) \to \text{CH}^*(F,*) }[/math] Showing the map in the reverse direction is an isomorphism is more work, but we get the isomorphisms [math]\displaystyle{ K_n^M(F) \to \text{CH}^n(F,n) }[/math] We can then relate the higher Chow groups to higher algebraic K-theory using the fact there are isomorphisms [math]\displaystyle{ K_n(X)\otimes \mathbb{Q} \cong \bigoplus_p \text{CH}^p(X,n)\otimes \mathbb{Q} }[/math] giving the relation to Quillen's higher algebraic K-theory. Note that the maps

[math]\displaystyle{ K^M_n(F) \to K_n(F) }[/math]

from the Milnor K-groups of a field to the Quillen K-groups, which is an isomorphism for [math]\displaystyle{ n\le 2 }[/math] but not for larger n, in general. For nonzero elements [math]\displaystyle{ a_1, \ldots, a_n }[/math] in F, the symbol [math]\displaystyle{ \{a_1, \ldots, a_n\} }[/math] in [math]\displaystyle{ K_n^M(F) }[/math] means the image of [math]\displaystyle{ a_1 \otimes \cdots \otimes a_n }[/math] in the tensor algebra. Every element of Milnor K-theory can be written as a finite sum of symbols. The fact that [math]\displaystyle{ \{a, 1-a\} = 0 }[/math] in [math]\displaystyle{ K_2^M(F) }[/math] for [math]\displaystyle{ a \in F\setminus \{0,1\} }[/math] is sometimes called the Steinberg relation.

Representation in motivic cohomology

In motivic cohomology, specifically motivic homotopy theory, there is a sheaf [math]\displaystyle{ K_{n,A} }[/math] representing a generalization of Milnor K-theory with coefficients in an abelian group [math]\displaystyle{ A }[/math]. If we denote [math]\displaystyle{ A_{tr}(X) = \mathbb{Z}_{tr}(X)\otimes A }[/math] then we define the sheaf [math]\displaystyle{ K_{n,A} }[/math] as the sheafification of the following pre-sheaf^{[5]pg 4} [math]\displaystyle{ K_{n,A}^{pre}: U \mapsto A_{tr}(\mathbb{A}^n)(U)/A_{tr}(\mathbb{A}^n - \{0\})(U) }[/math] Note that sections of this pre-sheaf are equivalent classes of cycles on [math]\displaystyle{ U\times\mathbb{A}^n }[/math] with coefficients in [math]\displaystyle{ A }[/math] which are equidimensional and finite over [math]\displaystyle{ U }[/math] (which follows straight from the definition of [math]\displaystyle{ \mathbb{Z}_{tr}(X) }[/math]). It can be shown there is an [math]\displaystyle{ \mathbb{A}^1 }[/math]-weak equivalence with the motivic Eilenberg-Maclane sheaves [math]\displaystyle{ K(A, 2n,n) }[/math] (depending on the grading convention).

Examples

Finite fields

For a finite field [math]\displaystyle{ F = \mathbb{F}_q }[/math], [math]\displaystyle{ K_1^M(F) }[/math] is a cyclic group of order [math]\displaystyle{ q-1 }[/math] (since is it isomorphic to [math]\displaystyle{ \mathbb{F}_q^* }[/math]), so graded commutativity gives [math]\displaystyle{ l(a)\cdot l(b) = -l(b)\cdot l(a) }[/math] hence [math]\displaystyle{ l(a)^2 =-l(a) ^2 }[/math] Because [math]\displaystyle{ K_2^M(F) }[/math] is a finite group, this implies it must have order [math]\displaystyle{ \leq 2 }[/math]. Looking further, [math]\displaystyle{ 1 }[/math] can always be expressed as a sum of quadratic non-residues, i.e. elements [math]\displaystyle{ a,b \in F }[/math] such that [math]\displaystyle{ [a],[b] \in F/F^{\times 2} }[/math] are not equal to [math]\displaystyle{ 0 }[/math], hence [math]\displaystyle{ a + b = 1 }[/math] showing [math]\displaystyle{ K_2^M(F) = 0 }[/math]. Because the Steinberg relations generate all relations in the Milnor K-theory ring, we have [math]\displaystyle{ K_n^M(F) = 0 }[/math] for [math]\displaystyle{ n \gt 2 }[/math].

Real numbers

For the field of real numbers [math]\displaystyle{ \mathbb{R} }[/math] the Milnor K-theory groups can be readily computed. In degree [math]\displaystyle{ n }[/math] the group is generated by [math]\displaystyle{ K_n^M(\mathbb{R}) = \{(-1)^n, l(a_1)\cdots l(a_n) : a_1,\ldots, a_n \gt 0 \} }[/math] where [math]\displaystyle{ (-1)^n }[/math] gives a group of order [math]\displaystyle{ 2 }[/math] and the subgroup generated by the [math]\displaystyle{ l(a_1)\cdots l(a_n) }[/math] is divisible. The subgroup generated by [math]\displaystyle{ (-1)^n }[/math] 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

[math]\displaystyle{ K^M_2(\Complex) }[/math] is an uncountable uniquely divisible group.^[7] Also, [math]\displaystyle{ K^M_2(\R) }[/math] is the direct sum of a cyclic group of order 2 and an uncountable uniquely divisible group; [math]\displaystyle{ K^M_2(\Q_p) }[/math] is the direct sum of the multiplicative group of [math]\displaystyle{ \mathbb{F}_p }[/math] and an uncountable uniquely divisible group; [math]\displaystyle{ K^M_2(\Q) }[/math] is the direct sum of the cyclic group of order 2 and cyclic groups of order [math]\displaystyle{ p-1 }[/math] for all odd prime [math]\displaystyle{ p }[/math]. For [math]\displaystyle{ n \geq 3 }[/math], [math]\displaystyle{ K_n^M(\mathbb{Q}) \cong \mathbb{Z}/2 }[/math]. 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 [math]\displaystyle{ K_2^M(F) }[/math] induced from the inclusion of a global field [math]\displaystyle{ F }[/math] to its completions [math]\displaystyle{ F_v }[/math], so there is a morphism[math]\displaystyle{ K_2^M(F) \to \bigoplus_{v} K_2^M(F_v)/(\text{max. divis. subgr.}) }[/math] whose kernel finitely generated. In addition, the cokernel is isomorphic to the roots of unity in [math]\displaystyle{ F }[/math].

In addition, for a general local field [math]\displaystyle{ F }[/math] (such as a finite extension [math]\displaystyle{ K/\mathbb{Q}_p }[/math]), the Milnor K-groups [math]\displaystyle{ K_n^M(F) }[/math] are divisible.

K_*^M(F(t))

There is a general structure theorem computing [math]\displaystyle{ K_n^M(F(t)) }[/math] for a field [math]\displaystyle{ F }[/math] in relation to the Milnor K-theory of [math]\displaystyle{ F }[/math] and extensions [math]\displaystyle{ F[t]/(\pi) }[/math] for non-zero primes ideals [math]\displaystyle{ (\pi) \in \text{Spec}(F[t]) }[/math]. This is given by an exact sequence [math]\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 }[/math] where [math]\displaystyle{ \partial_\pi : K_n^M(F(t)) \to K_{n-1}F[t]/(\pi) }[/math] is a morphism constructed from a reduction of [math]\displaystyle{ F }[/math] to [math]\displaystyle{ \overline{F}_v }[/math] for a discrete valuation [math]\displaystyle{ v }[/math]. This follows from the theorem there exists only one homomorphism [math]\displaystyle{ \partial:K_n^M(F) \to K_{n-1}^M(\overline{F}) }[/math] which for the group of units [math]\displaystyle{ U \subset F }[/math] which are elements have valuation [math]\displaystyle{ 0 }[/math], having a natural morphism [math]\displaystyle{ U \to \overline{F}_v^* }[/math] where [math]\displaystyle{ u \mapsto \overline{u} }[/math] we have [math]\displaystyle{ \partial(l(\pi)l(u_2)\cdots l(u_n)) = l(\overline{u}_2)\cdots l(\overline{u}_n) }[/math] where [math]\displaystyle{ \pi }[/math] a prime element, meaning [math]\displaystyle{ \text{Ord}_v(\pi) = 1 }[/math], and [math]\displaystyle{ \partial(l(u_1)\cdots l(u_n)) = 0 }[/math] Since every non-zero prime ideal [math]\displaystyle{ (\pi) \in \text{Spec}(F[t]) }[/math] gives a valuation [math]\displaystyle{ v_\pi : F(t) \to F[t]/(\pi) }[/math], we get the map [math]\displaystyle{ \partial_\pi }[/math] on the Milnor K-groups.

Applications

Milnor K-theory plays a fundamental role in higher class field theory, replacing [math]\displaystyle{ K_1^M(F) = F^{\times}\! }[/math] in the one-dimensional class field theory.

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

[math]\displaystyle{ K^M_n(F) \cong H^n(F, \Z(n)) }[/math]

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:

[math]\displaystyle{ K^M_n(F)/r \cong H^n_{\mathrm{et}}(F, \Z/r(n)), }[/math]

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 [math]\displaystyle{ n = 2 }[/math] and [math]\displaystyle{ r = 2 }[/math], 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 [math]\displaystyle{ W(F) \to\Z/2 }[/math] given by the dimension of a quadratic form, modulo 2. Milnor defined a homomorphism:

[math]\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} }[/math]

where [math]\displaystyle{ \langle \langle a_1, a_2, \ldots , a_n \rangle \rangle }[/math] 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 [math]\displaystyle{ K_n^M(F)/2 \to I^n/I^{n+1} }[/math] is an isomorphism.^[11]

See also

• Azumaya algebra
• Motivic homotopy theory

References

• Algebraic and geometric theory of quadratic forms, American Mathematical Society, 2008, ISBN 978-0-8218-4329-1 
• 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. 
• Lectures in Motivic Cohomology, Clay Mathematical Monographs, 2, American Mathematical Society, 2006, ISBN 978-0-8218-3847-1, http://math.rutgers.edu/~weibel/motiviclectures.html 
• Milnor, John Willard (1970), With an appendix by John Tate, "Algebraic K-theory and quadratic forms", Inventiones Mathematicae 9 (4): 318–344, doi:10.1007/BF01425486, ISSN 0020-9910, Bibcode: 1970InMat...9..318M 
• "An exact sequence for [math]\displaystyle{ K^M_*/2 }[/math] with applications to quadratic forms", Annals of Mathematics 165: 1–13, 2007, doi:10.4007/annals.2007.165.1 
• "On motivic cohomology with [math]\displaystyle{ \Z/\ell }[/math]-coefficients", Annals of Mathematics 174 (1): 401–438, 2011, doi:10.4007/annals.2011.174.1.11, http://annals.math.princeton.edu/2011/174-1/p11 

External links

• Some aspects of the functor [math]\displaystyle{ K 2 }[/math] of fields
• About Tate's computation of [math]\displaystyle{ K_2(\mathbb{Q}) }[/math]

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