Milnor K-theory


In mathematics, Milnor K-theory is an invariant of fields defined by . Originally viewed as an approximation to algebraic K-theory, Milnor K-theory has turned out to be an important invariant in its own right.

Definition

The calculation of K2 of a field by Hideya 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 by the two-sided ideal generated by:
The nth Milnor K-group is the nth graded piece of this graded ring; for example, and There is a natural homomorphism
from the Milnor K-groups of a field to the Daniel Quillen K-groups, which is an isomorphism for n ≤ 2 but not for larger n, in general. For nonzero elements in F, the symbol in 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 = 0 in for a in F − is sometimes called the Steinberg relation.
The ring is graded-commutative.

Examples

We have for n > 2, while is an uncountable uniquely divisible group. Also, is the direct sum of a cyclic group of order 2 and an uncountable uniquely divisible group; is the direct sum of the multiplicative group of and an uncountable uniquely divisible group; is the direct sum of the cyclic group of order 2 and cyclic groups of order for all odd prime.

Applications

Milnor K-theory plays a fundamental role in higher class field theory, replacing 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, relates Milnor K-theory to Galois cohomology or étale cohomology:
for any positive integer r invertible in the field F. This was proved by Vladimir Voevodsky, with contributions by Markus Rost and others. This includes the theorem of Alexander Merkurjev and Andrei Suslin and the Milnor conjecture as special cases.
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 given by the dimension of a quadratic form, modulo 2. Milnor defined a homomorphism:
where denotes the class of the n-fold Pfister form.
Orlov, Vishik, and Voevodsky proved another statement called the Milnor conjecture, namely that this homomorphism is an isomorphism.