Milnor K-theory

In mathematics, Milnor K-theory was an early attempt to define higher algebraic K-theory, introduced by Milnor (1970).

The calculation of K₂ of a field F led Milnor to the following ad hoc definition of "higher" K-groups by

thus as graded parts of a quotient of the tensor algebra of the multiplicative group F× by the two-sided ideal, generated by the

for a ≠ 0, 1. For n = 0,1,2 these coincide with Quillen's K-groups of a field, but for n ≧ 3 they differ in general. We define the symbol as the image of : the case n=2 is a Steinberg symbol.

For example, we have for n ≧ 2; is an uncountable uniquely divisible group; 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 .

Milnor K-theory plays a fundamental role in higher class field theory, replacing in the one dimensional class field theory.

Milnor K-theory modulo 2, denoted k_*(F) is related to étale (or Galois) cohomology of the field F by the Milnor conjecture, proven by Voevodsky. The analogous statement for odd primes is the Bloch–Kato conjecture, proved by Voevodsky, Rost, and others.

There are homomorphisms from k_n(F) to the Witt ring of F by taking the symbol

$\{a_1,\ldots,a_n\} \mapsto \langle \langle a_1, a_2, ..., a_n \rangle \rangle = \langle 1, a_1 \rangle \otimes \langle 1, a_2 \rangle \otimes ... \otimes \langle 1, a_n \rangle \ ,$

where the image is a Pfister form of dimension 2n. The image can be taken as In/In+1 and the map is surjective since the Pfister forms additively generate In. The Milnor conjecture can be interpreted as stating that these maps are isomorphisms.