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:
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 KnM(F) is the nth graded piece of this graded ring; for example, K0M(F) = Z and K1M(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, ..., an in F, the symbol {a1, ..., an} in KnM(F) 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 {a, 1−a} = 0 in K2M(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 for n ≧ 2, while is an uncountable uniquely divisible group. (An abelian group is uniquely divisible if it is a vector space over the rational numbers.) 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 K1M(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.[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:
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 KnM(F)/2 to the quotient In/In+1, sending a symbol {a1, ..., an} to the class of the n-fold Pfister form[4]
Orlov, Vishik, and Voevodsky proved another statement called the Milnor conjecture, namely that this homomorphism KnM(F)/2 → In/In+1 is an isomorphism.[5]
References
- 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