13.1 Milnor K-Theory and Bloch-Lichtenbaum spectral sequence
4 min read•july 30, 2024
simplifies , making it easier to compute. It's generated by symbols and follows specific relations, providing a more accessible way to understand the K-theory of fields.
The connects Milnor K-theory to algebraic K-theory and . This powerful tool helps calculate algebraic K-theory for various fields, revealing deep connections between these mathematical structures.
Milnor K-theory and algebraic K-theory
Definition and properties of Milnor K-theory
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Milnor K-theory, denoted KnM(F), is a graded ring associated to a field F
Generated by symbols {a1,...,an} for ai in F× subject to the Steinberg relations
Steinberg relations: multilinearity, {a,1−a}=1 for a=0,1, and {a,−a}=1
K0M(F) is defined as Z, K1M(F) is defined as F×, and higher KnM(F) are defined using the symbols and relations
Can be viewed as a "simplified" version of algebraic K-theory that is more computable and amenable to explicit calculations
Examples:
For F=Q, K1M(Q)=Q×={±1}×Z>0
For F=C, K2M(C) is generated by symbols {z1,z2} for z1,z2∈C×
Relationship between Milnor K-theory and algebraic K-theory
Natural map from Milnor K-theory to Quillen's algebraic K-theory, KnM(F)→Kn(F)
Sends the symbol {a1,...,an} to the product [a1]⋅...⋅[an] in Kn(F)
Isomorphism for n≤2 but not in general for higher n
Bloch-Lichtenbaum spectral sequence relates Milnor K-theory to the algebraic K-theory of fields
Provides a tool for computing algebraic K-theory using Milnor K-theory and étale cohomology
Examples:
For F=Q, the map K2M(Q)→K2(Q) is an isomorphism
For F=Fq(t), the map K3M(F)→K3(F) is not an isomorphism
Bloch-Lichtenbaum spectral sequence
Construction and properties of the Bloch-Lichtenbaum spectral sequence
Spectral sequence Erp,q converging to the algebraic K-theory of a field F
E2p,q=Hp(F,Z(q)) for p≤q and 0 otherwise, where Hp(F,Z(q)) is étale cohomology and Z(q) is the motivic complex
Differentials are motivic operations
Degenerates at E2 after tensoring with Q
Edge map E2n,n→Kn(F) coincides with the natural map from Milnor K-theory to algebraic K-theory
Provides a filtration on the algebraic K-theory of F, with the associated graded pieces expressed in terms of étale cohomology
Examples:
For F=Q, the spectral sequence degenerates at E2 and yields an exact sequence 0→Kn(Q)→KnM(Q)→⨁pHp(Q,Z(n−p))→0
For F=Fq(t), the spectral sequence also degenerates at E2 and provides information about the algebraic K-theory of F
Applications to computing algebraic K-theory of fields
Number fields: spectral sequence degenerates at E2 and yields an exact sequence
0→Kn(F)→KnM(F)→⨁pHp(F,Z(n−p))→0
Allows computation of Kn(F) in terms of Milnor K-theory and étale cohomology
Global fields of positive characteristic: similar degeneration results and exact sequences involving Milnor K-theory and étale cohomology
Local fields: spectral sequence does not always degenerate but still yields information about the structure of algebraic K-theory
Over p-adic fields, the differentials and extensions encode arithmetic information related to the Brauer group and local class field theory
Has been used to make progress on computing the algebraic K-theory of important classes of fields
Totally real number fields and function fields over finite fields
Examples:
For F=Q(5), the spectral sequence can be used to compute K3(F) in terms of K3M(F) and étale cohomology groups
For F=Qp, the spectral sequence provides information about the structure of Kn(Qp) and its relationship to local class field theory
Milnor K-theory vs étale cohomology
Connections between Milnor K-theory and étale cohomology
Bloch-Lichtenbaum spectral sequence expresses the algebraic K-theory of a field in terms of Milnor K-theory and étale cohomology
Reveals deep connections between these objects
Milnor K-theory of a field F can be identified with the étale cohomology group Hn(F,Z(n)) when F contains an algebraically closed field
Isomorphism known as the or the Totaro theorem
Natural map from Milnor K-theory to étale cohomology, KnM(F)→Hn(F,Z(n))
Isomorphism for n≤2 but not in general for higher n
Failure of this map to be an isomorphism is measured by the motivic cohomology groups Hp(F,Z(q)) for p=q, which appear in the Bloch-Lichtenbaum spectral sequence
Examples:
For F=C, the Nesterenko-Suslin theorem gives an isomorphism KnM(C)≅Hn(C,Z(n))
For F=Q, the map K3M(Q)→H3(Q,Z(3)) is not an isomorphism, and the difference is measured by motivic cohomology groups
Importance and research directions
Étale cohomology groups Hp(F,Z(q)) have a natural product structure
Corresponds to the product in Milnor K-theory under the isomorphism KnM(F)≅Hn(F,Z(n)) when F contains an algebraically closed field
Both Milnor K-theory and étale cohomology are important tools for studying the arithmetic and geometric properties of fields
Their relationship via the Bloch-Lichtenbaum spectral sequence has been the subject of extensive research
Motivic cohomology, which measures the difference between Milnor K-theory and étale cohomology, is an active area of study
Generalizations and analogues of the Bloch-Lichtenbaum spectral sequence, such as the and the , are also being investigated
Examples:
The , which relates Milnor K-theory to quadratic forms and , was a major open problem that was resolved using ideas from motivic cohomology and the Bloch-Lichtenbaum spectral sequence
The , which relates motivic cohomology to étale cohomology and provides a generalization of the Bloch-Lichtenbaum spectral sequence, is a central problem in the field that has been the subject of much recent work