In mathematics, Milnor K-theory[1] is an algebraic invariant (denoted for a field ) defined by John Milnor (1970) as an attempt to study higher algebraic K-theory in the special case of fields. It was hoped this would help illuminate the structure for algebraic K-theory and give some insight about its relationships with other parts of mathematics, such as Galois cohomology and the Grothendieck–Witt ring of quadratic forms. Before Milnor K-theory was defined, there existed ad-hoc definitions for and . Fortunately, it can be shown Milnor K-theory is a part of algebraic K-theory, which in general is the easiest part to compute.[2]
Definition[edit]
Motivation[edit]
After the definition of the Grothendieck group of a commutative ring, it was expected there should be an infinite set of invariants called higher K-theory groups, from the fact there exists a short exact sequence
which should have a continuation by a long exact sequence. Note the group on the left is relative K-theory. This led to much study and as a first guess for what this theory would look like, Milnor gave a definition for fields. His definition is based upon two calculations of what higher K-theory "should" look like in degrees and . Then, if in a later generalization of algebraic K-theory was given, if the generators of lived in degree and the relations in degree , then the constructions in degrees and would give the structure for the rest of the K-theory ring. Under this assumption, Milnor gave his "ad-hoc" definition. It turns out algebraic K-theory in general has a more complex structure, but for fields the Milnor K-theory groups are contained in the general algebraic K-theory groups after tensoring with , i.e. .[3] It turns out the natural map fails to be injective for a global field [3]pg 96.
Definition[edit]
Note for fields the Grothendieck group can be readily computed as since the only finitely generated modules are finite-dimensional vector spaces. Also, Milnor's definition of higher K-groups depends upon the canonical isomorphism
(the group of units of ) and observing the calculation of K2 of a field by Hideya Matsumoto, which gave the simple presentation
for a two-sided ideal generated by elements , called Steinberg relations. Milnor took the hypothesis that these were the only relations, hence he gave the following "ad-hoc" definition of Milnor K-theory as
The direct sum of these groups is isomorphic to a tensor algebra over the integers of the multiplicative group modded out by the two-sided ideal generated by:
so
showing his definition is a direct extension of the Steinberg relations.
Properties[edit]
Ring structure[edit]
The graded module is a graded-commutative ring[1]pg 1-3.[4] If we write
as
then for and we have
From the proof of this property, there are some additional properties which fall out, like
Relation to Higher Chow groups and Quillen's higher K-theory[edit]
One of the core properties relating Milnor K-theory to higher algebraic K-theory is the fact there exists natural isomorphisms
from the Milnor K-groups of a field to the Quillen K-groups, which is an isomorphism for but not for larger n, in general. For nonzero elements in F, the symbol in means the image of in the tensor algebra. Every element of Milnor K-theory can be written as a finite sum of symbols. The fact that in for is sometimes called the Steinberg relation.
Representation in motivic cohomology[edit]
In motivic cohomology, specifically motivic homotopy theory, there is a sheaf representing a generalization of Milnor K-theory with coefficients in an abelian group . If we denote then we define the sheaf as the sheafification of the following pre-sheaf[5]pg 4
Examples[edit]
Finite fields[edit]
For a finite field , is a cyclic group of order (since is it isomorphic to ), so graded commutativity gives
Real numbers[edit]
For the field of real numbers the Milnor K-theory groups can be readily computed. In degree the group is generated by
Other calculations[edit]
is an uncountable uniquely divisible group.[7] 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 . For , . The full proof is in the appendix of Milnor's original paper.[1] Some of the computation can be seen by looking at a map on induced from the inclusion of a global field to its completions , so there is a morphism
In addition, for a general local field (such as a finite extension ), the Milnor K-groups are divisible.
K*M(F(t))[edit]
There is a general structure theorem computing for a field in relation to the Milnor K-theory of and extensions for non-zero primes ideals . This is given by an exact sequence
Applications[edit]
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.[8] 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 étale cohomology:
for any positive integer r invertible in the field F. This conjecture was proved by Vladimir Voevodsky, with contributions by Markus Rost and others.[9] This includes the theorem of Alexander Merkurjev and Andrei Suslin as well as the Milnor conjecture as special cases (the cases when and , 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 given by the dimension of a quadratic form, modulo 2. Milnor defined a homomorphism:
where denotes the class of the n-fold Pfister form.[10]
Dmitri Orlov, Alexander Vishik, and Voevodsky proved another statement called the Milnor conjecture, namely that this homomorphism is an isomorphism.[11]
See also[edit]
References[edit]
- ^ a b c Milnor, John (1970-12-01). "Algebraic K -theory and quadratic forms". Inventiones Mathematicae. 9 (4): 318–344. Bibcode:1970InMat...9..318M. doi:10.1007/BF01425486. ISSN 1432-1297. S2CID 13549621.
- ^ a b Totaro, Burt. "Milnor K-Theory is the Simplest Part of Algebraic K-Theory" (PDF). Archived (PDF) from the original on 2 Dec 2020.
- ^ a b Shapiro, Jack M. (1981-01-01). "Relations between the milnor and quillen K-theory of fields". Journal of Pure and Applied Algebra. 20 (1): 93–102. doi:10.1016/0022-4049(81)90051-7. ISSN 0022-4049.
- ^ Gille & Szamuely (2006), p. 184.
- ^ Voevodsky, Vladimir (2001-07-15). "Reduced power operations in motivic cohomology". arXiv:math/0107109.
- ^ Bachmann, Tom (May 2018). "Motivic and Real Etale Stable Homotopy Theory". Compositio Mathematica. 154 (5): 883–917. arXiv:1608.08855. doi:10.1112/S0010437X17007710. ISSN 0010-437X. S2CID 119305101.
- ^ An abelian group is uniquely divisible if it is a vector space over the rational numbers.
- ^ Mazza, Voevodsky, Weibel (2005), Theorem 5.1.
- ^ Voevodsky (2011).
- ^ Elman, Karpenko, Merkurjev (2008), sections 5 and 9.B.
- ^ 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. Vol. 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), "Algebraic K-theory and quadratic forms", Inventiones Mathematicae, 9 (4), With an appendix by John Tate: 318–344, Bibcode:1970InMat...9..318M, doi:10.1007/BF01425486, ISSN 0020-9910, MR 0260844, S2CID 13549621, Zbl 0199.55501
- Orlov, Dmitri; Vishik, Alexander; Voevodsky, Vladimir (2007), "An exact sequence for with applications to quadratic forms", Annals of Mathematics, 165: 1–13, arXiv:math/0101023, doi:10.4007/annals.2007.165.1, MR 2276765, S2CID 9504456
- Voevodsky, Vladimir (2011), "On motivic cohomology with -coefficients", Annals of Mathematics, 174 (1): 401–438, arXiv:0805.4430, doi:10.4007/annals.2011.174.1.11, MR 2811603, S2CID 15583705