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In mathematics, Milnor K-theory is an algebraic invariant (denoted K_*(F) for a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
F) defined by as an attempt to study higher
algebraic K-theory Algebraic ''K''-theory is a subject area in mathematics with connections to geometry, topology, ring theory, and number theory. Geometric, algebraic, and arithmetic objects are assigned objects called ''K''-groups. These are groups in the sense ...
in the special case of
fields Fields may refer to: Music * Fields (band), an indie rock band formed in 2006 * Fields (progressive rock band), a progressive rock band formed in 1971 * ''Fields'' (album), an LP by Swedish-based indie rock band Junip (2010) * "Fields", a song b ...
. It was hoped this would help illuminate the structure for algebraic and give some insight about its relationships with other parts of mathematics, such as
Galois cohomology In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group ''G'' associated to a field extension ''L''/''K'' acts in a natur ...
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 is a part of algebraic , 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 groups, from the fact there exists a
short exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context ...
:K(R,I) \to K(R) \to K(R/I) \to 0 which should have a continuation by a
long exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the context ...
. Note the group on the left is relative . 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 "should" look like in degrees 1 and 2. Then, if in a later generalization of algebraic 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 ring. Under this assumption, Milnor gave his "ad-hoc" definition. It turns out algebraic K_*(R) in general has a more complex structure, but for fields the Milnor groups are contained in the general algebraic groups after tensoring with \mathbb, i.e. K^M_n(F)\otimes \mathbb \subseteq K_n(F)\otimes \mathbb. It turns out the natural map \lambda:K^M_4(F) \to K_4(F) fails to be injective for a global field Fpg 96.


Definition

Note for fields the Grothendieck group can be readily computed as K_0(F) = \mathbb since the only
finitely generated module In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring ''R'' may also be called a finite ''R''-module, finite over ''R'', or a module of finite type. Related concepts in ...
s are finite- dimensional
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s. Also, Milnor's definition of higher depends upon the canonical
isomorphism In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
:l\colon K_1(F) \to F^* (the group of units of F) and observing the calculation of ''K''2 of a field by
Hideya Matsumoto Hideya Matsumoto(英也, 松本)Vol.39 No.1 P.46 Journal of the Mathematical Society of Japan https://www.jstage.jst.go.jp/article/sugaku1947/39/1/39_1_43/_pdf/-char/ja is a Japanese mathematician who works on algebraic groups, who proved Matsu ...
, which gave the simple presentation :K_2(F) = \frac 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. The direct sum of these groups is isomorphic to a
tensor algebra In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of being ...
over the integers of the
multiplicative group In mathematics and group theory, the term multiplicative group refers to one of the following concepts: *the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referre ...
K_1(F) \cong F^* modded out by the
two-sided ideal In ring theory, a branch of abstract algebra, an ideal of a ring is a special subset of its elements. Ideals generalize certain subsets of the integers, such as the even numbers or the multiples of 3. Addition and subtraction of even numbers p ...
generated by: :\left \ so :\bigoplus_^\infty K_n^M(F) \cong \frac showing his definition is a direct extension of the Steinberg relations.


Properties


Ring structure

The graded module K_*^M(F) is a graded-commutative ringpg 1-3.Gille & Szamuely (2006), p. 184. 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^M_j(F) we have :\xi \cdot \eta = (-1)^\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 In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is b ...
every positive dimensional K_n^M(F) is nilpotent, which is a powerful statement about the structure of Milnor . In particular, for the fields \mathbb(i), \mathbb_p(i) with \sqrt \not\in \mathbb_p, all of its Milnor are nilpotent. In the converse case, the field F can be embedded into a
real closed field In mathematics, a real closed field is a field ''F'' that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers. D ...
, 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 K_n^M(F) \to \text^(F,n) to Bloch's Higher chow groups which induces a morphism of graded rings K_*^M(F) \to \text^*(F,*) This can be verified using an explicit morphismpg 181 \phi:F^* \to \text^1(F,1) where \phi(a)\phi(1-a) = 0 ~\text~ \text^2(F,2) ~\text~ a,1-a \in F^* This map is given by \begin \ &\mapsto 0 \in \text^1(F,1) \\ \ &\mapsto \in \text^1(F,1) \end for /math> the class of the point :1\in \mathbb^1_F-\ with a \in F^*-\. The main property to check is that + /a= 0 for a \in F^*-\ and + = b/math>. Note this is distinct from cdot /math> since this is an element in \text^2(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 C^1(F,2) whose image under the boundary map \partial is the sum + - b/math>for ab \neq 1, showing they differ by a boundary. Similarly, if ab=1 the boundary map sends this cycle to - /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 \text^p(F,q) \otimes \text^r(F,s) \to \text^(F,q+s) we get an explicit map K_*^M(F) \to \text^*(F,*) Showing the map in the reverse direction is an isomorphism is more work, but we get the isomorphisms K_n^M(F) \to \text^n(F,n) We can then relate the higher Chow groups to higher algebraic K-theory using the fact there are isomorphisms K_n(X)\otimes \mathbb \cong \bigoplus_p \text^p(X,n)\otimes \mathbb giving the relation to Quillen's higher algebraic K-theory. Note that the maps :K^M_n(F) \to K_n(F) from the Milnor K-groups of a field to the
Daniel Quillen Daniel Gray "Dan" Quillen (June 22, 1940 – April 30, 2011) was an American mathematician. He is known for being the "prime architect" of higher algebraic ''K''-theory, for which he was awarded the Cole Prize in 1975 and the Fields Medal in 197 ...
K-groups, which is an isomorphism for n\le 2 but not for larger ''n'', in general. For nonzero elements a_1, \ldots, a_n in ''F'', the symbol \ 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 \ = 0 in K_2^M(F) for a \in F\setminus \ is sometimes called the Steinberg relation.


Representation in motivic cohomology

In
motivic cohomology Motivic cohomology is an invariant of algebraic varieties and of more general schemes. It is a type of cohomology related to motives and includes the Chow ring of algebraic cycles as a special case. Some of the deepest problems in algebraic geo ...
, specifically
motivic homotopy theory In music, a motif IPA: ( /moʊˈtiːf/) (also motive) is a short musical phrase, a salient recurring figure, musical fragment or succession of notes that has some special importance in or is characteristic of a composition: "The motive ...
, there is a sheaf K_ representing a generalization of Milnor K-theory with coefficients in an abelian group A. If we denote A_(X) = \mathbb_(X)\otimes A then we define the sheaf K_ as the sheafification of the following pre-sheafpg 4 K_^: U \mapsto A_(\mathbb^n)(U)/A_(\mathbb^n - \)(U) Note that sections of this pre-sheaf are equivalent classes of cycles on U\times\mathbb^n with coefficients in A which are equidimensional and finite over U (which follows straight from the definition of \mathbb_(X)). It can be shown there is an \mathbb^1-weak equivalence with the motivic Eilenberg-Maclane sheaves K(A, 2n,n) (depending on the grading convention).


Examples


Finite fields

For a
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
F = \mathbb_q, K_1^M(F) is a cyclic group of order q-1 (since is it isomorphic to \mathbb_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 \in F/F^ 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 number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s \mathbb the Milnor groups can be readily computed. In degree n the group is generated by K_n^M(\mathbb) = \ 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^M_2(\Complex) is an
uncountable In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal num ...
uniquely divisible group. Also, K^M_2(\R) is the direct sum of a
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
of order 2 and an uncountable uniquely divisible group; K^M_2(\Q_p) is the direct sum of the multiplicative group of \mathbb_p and an uncountable uniquely divisible group; K^M_2(\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) \cong \mathbb/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_ K_2^M(F_v)/(\text) 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_p), the Milnor 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 of F and extensions F (\pi) for non-zero primes ideals (\pi) \in \text(F . This is given by an exact sequence 0 \to K_n^M(F) \to K_n^M(F(t)) \xrightarrow \bigoplus_ K_F (\pi) \to 0 where \partial_\pi : K_n^M(F(t)) \to K_F (\pi) is a morphism constructed from a reduction of F to \overline_v for a discrete valuation v. This follows from the theorem there exists only one homomorphism \partial:K_n^M(F) \to K_^M(\overline) which for the group of units U \subset F which are elements have valuation 0, having a natural morphism U \to \overline_v^* where u \mapsto \overline we have \partial(l(\pi)l(u_2)\cdots l(u_n)) = l(\overline_2)\cdots l(\overline_n) where \pi a prime element, meaning \text_v(\pi) = 1, and \partial(l(u_1)\cdots l(u_n)) = 0 Since every non-zero prime ideal (\pi) \in \text(F gives a valuation v_\pi : F(t) \to F (\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 In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local field, local and global field, global fields using objects associated to the ground f ...
, replacing K_1^M(F) = F^\! in the one-dimensional
class field theory In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field. Hilbert is credit ...
. Milnor K-theory fits into the broader context of
motivic cohomology Motivic cohomology is an invariant of algebraic varieties and of more general schemes. It is a type of cohomology related to motives and includes the Chow ring of algebraic cycles as a special case. Some of the deepest problems in algebraic geo ...
, via the isomorphism :K^M_n(F) \cong H^n(F, \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 becomes a theorem: certain motivic cohomology groups of a field can be explicitly computed by
generators and relations In mathematics, a presentation is one method of specifying a group. A presentation of a group ''G'' comprises a set ''S'' of generators—so that every element of the group can be written as a product of powers of some of these generators—and ...
. A much deeper result, the Bloch-Kato conjecture (also called the norm residue isomorphism theorem), relates Milnor to
Galois cohomology In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group ''G'' associated to a field extension ''L''/''K'' acts in a natur ...
or
étale cohomology In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjectur ...
: :K^M_n(F)/r \cong H^n_(F, \Z/r(n)), for any positive integer ''r'' invertible in the field ''F''. This conjecture was proved by
Vladimir Voevodsky Vladimir Alexandrovich Voevodsky (, russian: Влади́мир Алекса́ндрович Воево́дский; 4 June 1966 – 30 September 2017) was a Russian-American mathematician. His work in developing a homotopy theory for algebraic va ...
, with contributions by
Markus Rost Markus Rost is a German mathematician who works at the intersection of topology and algebra. He was an invited speaker at the International Congress of Mathematicians in 2002 in Beijing, China. He is a professor at the University of Bielefeld. He ...
and others. This includes the theorem of
Alexander Merkurjev Aleksandr Sergeyevich Merkurjev (russian: Алекса́ндр Сергее́вич Мерку́рьев, born September 25, 1955) is a Russian-American mathematician, who has made major contributions to the field of algebra. Currently Merkurjev ...
and
Andrei Suslin Andrei Suslin (russian: Андре́й Алекса́ндрович Су́слин, sometimes transliterated Souslin) was a Russian mathematician who contributed to algebraic K-theory and its connections with algebraic geometry. He was a Trustee ...
as well as the
Milnor conjecture In mathematics, the Milnor conjecture was a proposal by of a description of the Milnor K-theory (mod 2) of a general field ''F'' with characteristic different from 2, by means of the Galois (or equivalently étale) cohomology of ''F'' wi ...
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\Z/2 given by the dimension of a quadratic form, modulo 2. Milnor defined a homomorphism: :\begin K_n^M(F)/2 \to I^n/I^ \\ \ \mapsto \langle \langle a_1, \ldots , a_n \rangle \rangle = \langle 1, -a_1 \rangle \otimes \cdots \otimes \langle 1, -a_n \rangle \end where \langle \langle a_1, a_2, \ldots , a_n \rangle \rangle denotes the class of the ''n''-fold
Pfister form In mathematics, a Pfister form is a particular kind of quadratic form, introduced by Albrecht Pfister in 1965. In what follows, quadratic forms are considered over a field ''F'' of characteristic not 2. For a natural number ''n'', an ''n''-fold P ...
. 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^ is an isomorphism.Orlov, Vishik, Voevodsky (2007).


See also

*
Azumaya algebra In mathematics, an Azumaya algebra is a generalization of central simple algebras to ''R''-algebras where ''R'' need not be a field. Such a notion was introduced in a 1951 paper of Goro Azumaya, for the case where ''R'' is a commutative local rin ...
*
Motivic homotopy theory In music, a motif IPA: ( /moʊˈtiːf/) (also motive) is a short musical phrase, a salient recurring figure, musical fragment or succession of notes that has some special importance in or is characteristic of a composition: "The motive ...


References

* * * * * *


External links

* Some aspects of the functor K_2 of fields
About Tate's computation of K_2(\mathbb{Q})
K-theory