Milnor K-theory

Short description: Algebraic invariant

In mathematics, Milnor K-theory[1] 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 K1 and K2. 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]

## Contents

* 1 Definition
* 1.1 Motivation
* 1.2 Definition
* 2 Properties
* 2.1 Ring structure
* 2.2 Relation to Higher Chow groups and Quillen's higher K-theory
* 2.3 Representation in motivic cohomology
* 3 Examples
* 3.1 Finite fields
* 3.2 Real numbers
* 3.3 Other calculations
* 3.4 K*M(F(t))
* 4 Applications
* 5 See also
* 6 References
* 7 External links

## Definition

### Motivation

After the definition of the Grothendieck group K(R) of a commutative ring, it was expected there should be a sequence of invariants Ki(R) called higher K-theory groups, from the fact that there exists a short exact sequence

K(R,I)->K(R)->K(R/I)->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 1 and 2. Then, if in a later generalization of algebraic K-theory was given, if the generators of K*(R) lived in degree 1 and the relations in degree 2, then the constructions in degrees 1 and 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 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 ℚ, i.e. KnM(F)⊗ℚ⊆Kn(F)⊗ℚ.[3] It turns out the natural map λ:K4M(F)->K4(F) fails to be injective for a global field F[3]pg 96.

### Definition

Note for fields the Grothendieck group can be readily computed as K0(F)=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:K1(F)->F*

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

K2(F)=F*⊗F*{l(a)⊗l(1−a):a≠0,1}

for a two-sided ideal generated by elements l(a)⊗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

KnM(F)=K1(F)⊗⋯⊗K1(F){l(a1)⊗⋯⊗l(an):ai+ai+1=1}.

The direct sum of these groups is isomorphic to a tensor algebra over the integers of the multiplicative group K1(F)≅F* modded out by the two-sided ideal generated by:

{l(a)⊗l(1−a):0,1≠a∈F}

⨁n=0∞KnM(F)≅T*(K1M(F)){l(a)⊗l(1−a):a≠0,1}

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

## Properties

### Ring structure

The graded module K*M(F) is a graded-commutative ring[1]pg 1-3.[4] If we write

(l(a1)⊗⋯⊗l(an))⋅(l(b1)⊗⋯⊗l(bm))

l(a1)⊗⋯⊗l(an)⊗l(b1)⊗⋯⊗l(bm)

then for ξ∈KiM(F) and η∈KjM(F) we have

ξ⋅η=(−1)i⋅jη⋅ξ.

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)∈K1(F) since l(a)l(−a)=0. Also, if a1+⋯+an of non-zero fields elements equals 0,1, then l(a1)⋯l(an)=0 There's a direct arithmetic application: −1∈F is a sum of squares if and only if every positive dimensional KnM(F) is nilpotent, which is a powerful statement about the structure of Milnor K-groups. In particular, for the fields ℚ(i), ℚp(i) with −1∉ℚp, 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 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 KnM(F)->CHn(F,n) to Bloch's Higher chow groups which induces a morphism of graded rings K*M(F)->CH*(F,*) This can be verified using an explicit morphism[2]pg 181 ϕ:F*->CH1(F,1) where ϕ(a)ϕ(1−a)=0inCH2(F,2)fora,1−a∈F* This map is given by {1}↦0∈CH1(F,1){a}↦[a]∈CH1(F,1) for [a] the class of the point [a:1]∈ℙF1−{0,1,∞} with a∈F*−{1}. The main property to check is that [a]+[1/a]=0 for a∈F*−{1} and [a]+[b]=[ab]. Note this is distinct from [a]⋅[b] since this is an element in CH2(F,2). Also, the second property implies the first for b=1/a. This check can be done using a rational curve defining a cycle in C1(F,2) whose image under the boundary map ∂ is the sum [a]+[b]−[ab]for ab≠1, showing they differ by a boundary. Similarly, if ab=1 the boundary map sends this cycle to [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 CHp(F,q)⊗CHr(F,s)->CHp+r(F,q+s) we get an explicit map K*M(F)->CH*(F,*) Showing the map in the reverse direction is an isomorphism is more work, but we get the isomorphisms KnM(F)->CHn(F,n) We can then relate the higher Chow groups to higher algebraic K-theory using the fact there are isomorphisms Kn(X)⊗ℚ≅⨁pCHp(X,n)⊗ℚ giving the relation to Quillen's higher algebraic K-theory. Note that the maps

KnM(F)->Kn(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∈F∖{0,1} is sometimes called the Steinberg relation.

### Representation in motivic cohomology

In motivic cohomology, specifically motivic homotopy theory, there is a sheaf Kn,A representing a generalization of Milnor K-theory with coefficients in an abelian group A. If we denote Atr(X)=ℤtr(X)⊗A then we define the sheaf Kn,A as the sheafification of the following pre-sheaf[5]pg 4 Kn,Apre:U↦Atr(𝔸n)(U)/Atr(𝔸n−{0})(U) Note that sections of this pre-sheaf are equivalent classes of cycles on U×𝔸n with coefficients in A which are equidimensional and finite over U (which follows straight from the definition of ℤtr(X)). It can be shown there is an 𝔸1-weak equivalence with the motivic Eilenberg-Maclane sheaves K(A,2n,n) (depending on the grading convention).

## Examples

### Finite fields

For a finite field F=𝔽q, K1M(F) is a cyclic group of order q−1 (since is it isomorphic to 𝔽q*), so graded commutativity gives l(a)⋅l(b)=−l(b)⋅l(a) hence l(a)2=−l(a)2 Because K2M(F) is a finite group, this implies it must have order ≤2. Looking further, 1 can always be expressed as a sum of quadratic non-residues, i.e. elements a,b∈F such that [a],[b]∈F/F×2 are not equal to 0, hence a+b=1 showing K2M(F)=0. Because the Steinberg relations generate all relations in the Milnor K-theory ring, we have KnM(F)=0 for n>2.

### Real numbers

For the field of real numbers ℝ the Milnor K-theory groups can be readily computed. In degree n the group is generated by KnM(ℝ)={(−1)n,l(a1)⋯l(an):a1,…,an>0} where (−1)n gives a group of order 2 and the subgroup generated by the l(a1)⋯l(an) 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.[6] The others are lifts from the classical Steenrod operations to motivic cohomology.

### Other calculations

K2M(ℂ) is an uncountable uniquely divisible group.[7] Also, K2M(ℝ) is the direct sum of a cyclic group of order 2 and an uncountable uniquely divisible group; K2M(ℚp) is the direct sum of the multiplicative group of 𝔽p and an uncountable uniquely divisible group; K2M(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≥3, KnM(ℚ)≅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 K2M(F) induced from the inclusion of a global field F to its completions Fv, so there is a morphismK2M(F)->⨁vK2M(Fv)/(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/ℚp), the Milnor K-groups KnM(F) are divisible.

### K*M(F(t))

There is a general structure theorem computing KnM(F(t)) for a field F in relation to the Milnor K-theory of F and extensions F[t]/(π) for non-zero primes ideals (π)∈Spec(F[t]). This is given by an exact sequence 0->KnM(F)->KnM(F(t))→∂π⨁(π)∈Spec(F[t])Kn−1F[t]/(π)->0 where ∂π:KnM(F(t))->Kn−1F[t]/(π) is a morphism constructed from a reduction of F to F‾v for a discrete valuation v. This follows from the theorem there exists only one homomorphism ∂:KnM(F)->Kn−1M(F‾) which for the group of units U⊂F which are elements have valuation 0, having a natural morphism U->F‾v* where u↦u‾ we have ∂(l(π)l(u2)⋯l(un))=l(u‾2)⋯l(u‾n) where π a prime element, meaning Ordv(π)=1, and ∂(l(u1)⋯l(un))=0 Since every non-zero prime ideal (π)∈Spec(F[t]) gives a valuation vπ:F(t)->F[t]/(π), we get the map ∂π on the Milnor K-groups.

## 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

KnM(F)≅Hn(F,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:

KnM(F)/r≅Hetn(F,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 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:

{KnM(F)/2->In/In+1{a1,…,an}↦⟨⟨a1,…,an⟩⟩=⟨1,−a1⟩⊗⋯⊗⟨1,−an⟩

where ⟨⟨a1,a2,…,an⟩⟩ 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 KnM(F)/2->In/In+1 is an isomorphism.[11]

## See also

* Azumaya algebra
* Motivic homotopy theory

## References

1. ↑ 1.0 1.1 1.2 Milnor, John (1970-12-01). [330} "Algebraic K -theory and quadratic forms"] (in en). Inventiones Mathematicae 9 (4): 318–344. doi:10.1007/BF01425486. ISSN 1432-1297. Bibcode: 1970InMat...9..318M. https://gdz.sub.uni-goettingen.de/id/PPN356556735_0009?tify={"pages":[330]}.
2. ↑ 2.0 2.1 Totaro, Burt. "Milnor K-Theory is the Simplest Part of Algebraic K-Theory". https://www.math.ucla.edu/~totaro/papers/public_html/milnor.pdf.
3. ↑ 3.0 3.1 Shapiro, Jack M. (1981-01-01). "Relations between the milnor and quillen K-theory of fields" (in en). Journal of Pure and Applied Algebra 20 (1): 93–102. doi:10.1016/0022-4049(81)90051-7. ISSN 0022-4049.
4. ↑ Gille & Szamuely (2006), p. 184.
5. ↑ Voevodsky, Vladimir (2001-07-15). "Reduced power operations in motivic cohomology". arXiv:math/0107109.
6. ↑ Bachmann, Tom (May 2018). "Motivic and Real Etale Stable Homotopy Theory". Compositio Mathematica 154 (5): 883–917. doi:10.1112/S0010437X17007710. ISSN 0010-437X.
7. ↑ An abelian group is uniquely divisible if it is a vector space over the rational numbers.
8. ↑ Mazza, Voevodsky, Weibel (2005), Theorem 5.1.
9. ↑ Voevodsky (2011).
10. ↑ Elman, Karpenko, Merkurjev (2008), sections 5 and 9.B.
11. ↑ Orlov, Vishik, Voevodsky (2007).

* 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 K*M/2 with applications to quadratic forms", Annals of Mathematics 165: 1–13, 2007, doi:10.4007/annals.2007.165.1
* "On motivic cohomology with ℤ/ℓ-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 K2 of fields
* About Tate's computation of K2(ℚ)

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