Milnor K-theory

In mathematics, Milnor K-theory is an invariant of fields defined by Milnor (1970). 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 Matsumoto led Milnor to the following, seemingly naive, definition of the "higher" K-groups of a field F:

$K^M_*(F) := T^*F^\times/(a\otimes (1-a)), \,$

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

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

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

$K^M_n(F) \rightarrow K_n(F)$

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.^[1]

Examples

We have $K^M_n(\mathbb{F}_q) = 0$ for n ≧ 2, while $K^M_2(\mathbb{C})$ is an uncountable uniquely divisible group. (An abelian group is uniquely divisible if it is a vector space over the rational numbers.) Also, $K^M_2(\mathbb{R})$ is the direct sum of a cyclic group of order 2 and an uncountable uniquely divisible group; $K^M_2(\mathbb{Q}_p)$ is the direct sum of the multiplicative group of $\mathbb{F}_p$ and an uncountable uniquely divisible group; $K^M_2(\mathbb{Q})$ 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 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

$K_M^n(F) \cong H^n(F, \mathbf{Z}(n))$

of the Milnor K-theory of a field with a certain motivic cohomology group.^[2] 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:

$K_M^n(F)/r \cong H^n_{\text{et}}(F, \mathbf{Z}/r(n)),$

for any positive integer r invertible in the field F. This was proved by Voevodsky, with contributions by Rost and others.^[3] 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^[4]

$\langle \langle a_1, a_2, ... , a_n \rangle \rangle = \langle 1, -a_1 \rangle \otimes \langle 1, -a_2 \rangle \otimes ... \otimes \langle 1, -a_n \rangle.$

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.^[5]

References

1. Gille & Szamuely (2006), p. 184.
2. Mazza, Voevodsky, Weibel (2005), Theorem 5.1.
3. Voevodsky (2011).
4. Elman, Karpenko, Merkurjev (2008), sections 5 and 9.B.
5. Orlov, Vishik, Voevodsky (2007).

• Elman, Richard; Karpenko, Nikita; Merkurjev, Alexander (2008), Algebraic and geometric theory of quadratic forms, American Mathematical Society, ISBN 978-0-8218-4329-1, MR 2427530
• Gille, Philippe; Szamuely, Tamás (2006). Central simple algebras and Galois cohomology. Cambridge Studies in Advanced Mathematics 101. Cambridge: Cambridge University Press. ISBN 0-521-86103-9. MR 2266528. Zbl 1137.12001.
• Mazza, Carlo; Voevodsky, Vladimir; Weibel, Charles (2006), Lectures in Motivic Cohomology, Clay Mathematical Monographs, vol. 2, American Mathematical Society, ISBN 978-0-8218-3847-1, MR 2242284
• Milnor, John Willard (1970), With an appendix by J. Tate, "Algebraic K-theory and quadratic forms", Inventiones Mathematicae 9: 318–344, doi:10.1007/BF01425486, ISSN 0020-9910, MR 0260844, Zbl 0199.55501
• Orlov, Dmitri; Vishik, Alexander; Voevodsky, Vladimir (2007), "An exact sequence for K_*^M/2 with applications to quadratic forms", Annals of Mathematics 165: 1–13, doi:10.4007/annals.2007.165.1, MR 2276765
• Voevodsky, Vladimir (2011), "On motivic cohomology with Z/l-coefficients", Annals of Mathematics 174 (1): 401–438, doi:10.4007/annals.2011.174.1.11, MR 2811603