Milnor K-theory

In mathematics, Milnor K-theory is an algebraic invariant (denoted formula_1 for a field formula_2) defined by 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 formula_3 and formula_4. 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 formula_5 of a commutative ring, it was expected there should be an infinite set of invariants formula_6 called higher K-theory groups, from the fact there exists a short exact sequence
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. His definition is based upon two calculations of what higher K-theory "should" look like in degrees formula_8 and formula_9. Then, if in a later generalization of algebraic K-theory was given, if the generators of formula_10 lived in degree formula_8 and the relations in degree formula_9, then the constructions in degrees formula_8 and formula_9 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 formula_10 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 formula_16, i.e. formula_17. It turns out the natural map formula_18 fails to be injective for a global field formula_2pg 96.
Definition.
Note for fields the Grothendieck group can be readily computed as formula_20 since the only finitely generated modules are finite dimensional vector spaces. Also, Milnor's definition of higher K-groups depends upon the canonical isomorphism
(the group of units of formula_2) and observing the calculation of "K"2 of a field by Hideya Matsumoto, which gave the simple presentation
for a two-sided ideal generated by elements formula_24, 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
The direct sum of these groups is isomorphic to a tensor algebra over the integers of the multiplicative group formula_26 modded out by the two-sided ideal generated by:
so
showing his definition is a direct extension of the Steinberg relations.
Properties.
Ring structure.
The graded module formula_29 is a graded-commutative ringpg 1-3. If we write
as
then for formula_32 and formula_33 we have
From the proof of this property, there are some additional properties which fall out, like formula_35 for formula_36 since formula_37. Also, if formula_38 of non-zero fields elements equals formula_39, then formula_40 There's a direct arithmetic application: formula_41 is a sum of squares if and only if every positive dimensional formula_42 is nilpotent, which is a powerful statement about the structure of Milnor K-groups. In particular, for the fields formula_43, formula_44 with formula_45, all of its Milnor K-groups are nilpotent. In the converse case, the field formula_46 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 formula_47 to Bloch's Higher chow groups which induces a morphism of graded rings formula_48 This can be verified using an explicit morphismpg 181 formula_49 where formula_50 This map is given by formula_51 for formula_52 the class of the point formula_53 with formula_54. The main property to check is that formula_55 for formula_54 and formula_57. Note this is distinct from formula_58 since this is an element in formula_59. Also, the second property implies the first for formula_60. This check can be done using a rational curve defining a cycle in formula_61 whose image under the boundary map formula_62 is the sum formula_63for formula_64, showing they differ by a boundary. Similarly, if formula_65 the boundary map sends this cycle to formula_66, 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 formula_67 we get an explicit map formula_48 Showing the map in the reverse direction is an isomorphism is more work, but we get the isomorphisms formula_69 We can then relate the higher Chow groups to higher algebraic K-theory using the fact there are isomorphisms formula_70 giving the relation to Quillen's higher algebraic K-theory. Note that the maps
from the Milnor K-groups of a field to the Daniel Quillen K-groups, which is an isomorphism for formula_72 but not for larger "n", in general. For nonzero elements formula_73 in "F", the symbol formula_74 in formula_75 means the image of formula_76 in the tensor algebra. Every element of Milnor K-theory can be written as a finite sum of symbols. The fact that formula_77 in formula_78 for formula_79 is sometimes called the Steinberg relation.
Representation in motivic cohomology.
In motivic cohomology, specifically motivic homotopy theory, there is a sheaf formula_80 representing a generalization of Milnor K-theory with coefficients in an abelian group formula_81. If we denote formula_82 then we define the sheaf formula_80 as the sheafification of the following pre-sheafpg 4 formula_84 Note that sections of this pre-sheaf are equivalent classes of cycles on formula_85 with coefficients in formula_81 which are equidimensional and finite over formula_87 (which follows straight from the definition of formula_88). It can be shown there is an formula_89-weak equivalence with the motivic Eilenberg-Maclane sheaves formula_90 (depending on the grading convention).
Examples.
Finite fields.
For a finite field formula_91, formula_92 is a cyclic group of order formula_93 (since is it isomorphic to formula_94), so graded commutativity gives formula_95 hence formula_96 Because formula_97 is a finite group, this implies it must have order formula_98. Looking further, formula_99 can always be expressed as a sum of quadratic non-residues, i.e. elements formula_100 such that formula_101 are not equal to formula_102, hence formula_103 showing formula_104. Because the Steinberg relations generate all relations in the Milnor K-theory ring, we have formula_105 for formula_106.
Real numbers.
For the field of real numbers formula_107 the Milnor K-theory groups can be readily computed. In degree formula_108 the group is generated by formula_109 where formula_110 gives a group of order formula_111 and the subgroup generated by the formula_112 is divisible. The subgroup generated by formula_110 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.
formula_114 is an uncountable uniquely divisible group. Also, formula_115 is the direct sum of a cyclic group of order 2 and an uncountable uniquely divisible group; formula_116 is the direct sum of the multiplicative group of formula_117 and an uncountable uniquely divisible group; formula_118 is the direct sum of the cyclic group of order 2 and cyclic groups of order formula_119 for all odd prime formula_120. For formula_121, formula_122. 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 formula_97 induced from the inclusion of a global field formula_46 to its completions formula_125, so there is a morphismformula_126 whose kernel finitely generated. In addition, the cokernel is isomorphic to the roots of unity in formula_46.
In addition, for a general local field formula_46 (such as a finite extension formula_129), the Milnor K-groups formula_42 are divisible.
K*M(F(t)).
There is a general structure theorem computing formula_131 for a field formula_46 in relation to the Milnor K-theory of formula_46 and extensions formula_134 for non-zero primes ideals formula_135. This is given by an exact sequence formula_136 where formula_137 is a morphism constructed from a reduction of formula_46 to formula_139 for a discrete valuation formula_140. This follows from the theorem there exists only one homomorphism formula_141 which for the group of units formula_142 which are elements have valuation formula_102, having a natural morphism formula_144 where formula_145 we have formula_146 where formula_147 a prime element, meaning formula_148, and formula_149 Since every non-zero prime ideal formula_135 gives a valuation formula_151, we get the map formula_152 on the Milnor K-groups.
Applications.
Milnor K-theory plays a fundamental role in higher class field theory, replacing formula_153 in the one-dimensional class field theory.
Milnor K-theory fits into the broader context of motivic cohomology, via the isomorphism
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:
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 formula_156 and formula_157, 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 formula_158 given by the dimension of a quadratic form, modulo 2. Milnor defined a homomorphism:
where formula_160 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 formula_161 is an isomorphism.