Milnor K theory

Definition

The calculation of K₂ of a field by Matsumoto led Milnor to the following, seemingly naive, definition of the "higher" K-groups of a field F:

the quotient of the tensor algebra over the integers of the multiplicative group F^× by the two-sided ideal generated by the elements

for a ≠ 0, 1 in F. The nth Milnor K-group K_n^M(F) is the nth graded piece of this graded ring; for example, K₀^M(F) = Z and K₁^M(F) = F*. There is a natural homomorphism

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, ..., a_n in F, the symbol {a₁, ..., a_n} in K_n^M(F) means the image of a₁ ⊗ ... ⊗ 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 K₂^M(F) for a in F − {0,1} is sometimes called the Steinberg relation.

The ring K_*^M(F) is graded-commutative.

Examples

We have K n M ( F q ) = 0 for n ≧ 2, while K 2 M ( C ) is an uncountable uniquely divisible group. (An abelian group is uniquely divisible if it is a vector space over the rational numbers.) Also, K 2 M ( R ) is the direct sum of a cyclic group of order 2 and an uncountable uniquely divisible group; K 2 M ( Q p ) is the direct sum of the multiplicative group of F p and an uncountable uniquely divisible group; K 2 M ( Q ) is the direct sum of the cyclic group of order 2 and cyclic groups of order p − 1 for all odd prime p .

Applications

Milnor K-theory plays a fundamental role in higher class field theory, replacing K₁^M(F) = F* 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 etale cohomology:

for any positive integer r invertible in the field F. This was proved by Voevodsky, with contributions by Rost and others. This includes the Merkurjev−Suslin theorem and the Milnor conjecture as special cases (the cases 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 from the mod 2 Milnor K-group K_n^M(F)/2 to the quotient Iⁿ/Iⁿ⁺¹, sending a symbol {a₁, ..., a_n} to the class of the n-fold Pfister form

Orlov, Vishik, and Voevodsky proved another statement called the Milnor conjecture, namely that this homomorphism K_n^M(F)/2 → Iⁿ/Iⁿ⁺¹ is an isomorphism.