Milnor K-theory

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

Definition

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

$K_{*}^{M}(F):=T^{*}F^{\times }/(a\otimes (1-a)),\,$

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

$a\otimes (1-a)\,$

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 $\{a_{1},\ldots ,a_{n}\}$ as the image of $a_{1}\otimes \cdots \otimes a_{n}$: the case n=2 is a Steinberg symbol.

The tensor product on the tensor algebra induces a product $K_{m}\times K_{n}\rightarrow K_{m+n}$ making $K_{*}^{M}(F)$ a graded ring which is graded-commutative.

Examples

For example, we have $K_{n}^{M}(\mathbb {F} _{q})=0$ for n ≧ 2; $K_{2}^{M}(\mathbb {C} )$ is an uncountable uniquely divisible group; $K_{2}^{M}(\mathbb {R} )$ is the direct sum of a cyclic group of order 2 and an uncountable uniquely divisible group; $K_{2}^{M}(\mathbb {Q} _{p})$ is the direct sum of the multiplicative group of $\mathbb {F} _{p}$ and an uncountable uniquely divisible group; $K_{2}^{M}(\mathbb {Q} )$ 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 $K_{1}^{M}$ 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.

References

• Gille, Philippe; Szamuely, Tamás (2006). Central simple algebras and Galois cohomology. Cambridge Studies in Advanced Mathematics. Vol. 101. Cambridge: Cambridge University Press. ISBN 0-521-86103-9. Zbl 1137.12001.
• Lam, Tsit-Yuen (2005). Introduction to Quadratic Forms over Fields. Graduate Studies in Mathematics. Vol. 67. American Mathematical Society. ISBN 0-8218-1095-2. MR 2104929. Zbl 1068.11023.
• Milnor, John Willard (1970), "Algebraic K-theory and quadratic forms", Inventiones Mathematicae, 9, With an appendix by J. Tate: 318–344, doi:10.1007/BF01425486, ISSN 0020-9910, MR 0260844, Zbl 0199.55501