13.1 Milnor K-Theory and Bloch-Lichtenbaum spectral sequence

Milnor K-theory simplifies algebraic K-theory, 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 Bloch-Lichtenbaum spectral sequence connects Milnor K-theory to algebraic K-theory and étale cohomology. 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

Top images from around the web for Definition and properties of Milnor K-theory

• 

  Associated graded ring - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  Associated graded ring - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  Associated graded ring - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  Associated graded ring - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  Associated graded ring - Wikipedia, the free encyclopedia View original

  Is this image relevant?

1 of 3

Top images from around the web for Definition and properties of Milnor K-theory

• 

  Associated graded ring - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  Associated graded ring - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  Associated graded ring - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  Associated graded ring - Wikipedia, the free encyclopedia View original

  Is this image relevant?

• 

  Associated graded ring - Wikipedia, the free encyclopedia View original

  Is this image relevant?

1 of 3

• Milnor K-theory, denoted $K_n^M(F)$, is a graded ring associated to a field $F$
  • Generated by symbols $\{a_1, ..., a_n\}$ for $a_i$ in $F^×$ subject to the Steinberg relations
    • Steinberg relations: multilinearity, $\{a, 1-a\} = 1$ for $a \neq 0,1$, and $\{a, -a\} = 1$
  • $K_0^M(F)$ is defined as $\mathbb{Z}$, $K_1^M(F)$ is defined as $F^×$, and higher $K_n^M(F)$ 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 $F = \mathbb{Q}$, $K_1^M(\mathbb{Q}) = \mathbb{Q}^× = \{\pm 1\} \times \mathbb{Z}_{>0}$
  • For $F = \mathbb{C}$, $K_2^M(\mathbb{C})$ is generated by symbols $\{z_1, z_2\}$ for $z_1, z_2 \in \mathbb{C}^×$

Relationship between Milnor K-theory and algebraic K-theory

• Natural map from Milnor K-theory to Quillen's algebraic K-theory, $K_n^M(F) \to K_n(F)$
  • Sends the symbol $\{a_1, ..., a_n\}$ to the product $[a_1] \cdot ... \cdot [a_n]$ in $K_n(F)$
  • Isomorphism for $n \leq 2$ 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 $F = \mathbb{Q}$, the map $K_2^M(\mathbb{Q}) \to K_2(\mathbb{Q})$ is an isomorphism
  • For $F = \mathbb{F}_q(t)$, the map $K_3^M(F) \to K_3(F)$ is not an isomorphism

Bloch-Lichtenbaum spectral sequence

Construction and properties of the Bloch-Lichtenbaum spectral sequence

• Spectral sequence $E_r^{p,q}$ converging to the algebraic K-theory of a field $F$
  • $E_2^{p,q} = H^p(F, \mathbb{Z}(q))$ for $p \leq q$ and 0 otherwise, where $H^p(F, \mathbb{Z}(q))$ is étale cohomology and $\mathbb{Z}(q)$ is the motivic complex
  • Differentials are motivic operations
  • Degenerates at $E_2$ after tensoring with $\mathbb{Q}$
• Edge map $E_2^{n,n} \to K_n(F)$ 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 $F = \mathbb{Q}$, the spectral sequence degenerates at $E_2$ and yields an exact sequence $0 \to K_n(\mathbb{Q}) \to K_n^M(\mathbb{Q}) \to \bigoplus_p H^p(\mathbb{Q}, \mathbb{Z}(n-p)) \to 0$
  • For $F = \mathbb{F}_q(t)$, the spectral sequence also degenerates at $E_2$ and provides information about the algebraic K-theory of $F$

Applications to computing algebraic K-theory of fields

• Number fields: spectral sequence degenerates at $E_2$ and yields an exact sequence
  • $0 \to K_n(F) \to K_n^M(F) \to \bigoplus_p H^p(F, \mathbb{Z}(n-p)) \to 0$
  • Allows computation of $K_n(F)$ 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 $F = \mathbb{Q}(\sqrt{5})$, the spectral sequence can be used to compute $K_3(F)$ in terms of $K_3^M(F)$ and étale cohomology groups
  • For $F = \mathbb{Q}_p$, the spectral sequence provides information about the structure of $K_n(\mathbb{Q}_p)$ 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 $H^n(F, \mathbb{Z}(n))$ when $F$ contains an algebraically closed field
  • Isomorphism known as the Nesterenko-Suslin theorem or the Totaro theorem
• Natural map from Milnor K-theory to étale cohomology, $K_n^M(F) \to H^n(F, \mathbb{Z}(n))$
  • Isomorphism for $n \leq 2$ but not in general for higher $n$
  • Failure of this map to be an isomorphism is measured by the motivic cohomology groups $H^p(F, \mathbb{Z}(q))$ for $p \neq q$, which appear in the Bloch-Lichtenbaum spectral sequence
• Examples:
  • For $F = \mathbb{C}$, the Nesterenko-Suslin theorem gives an isomorphism $K_n^M(\mathbb{C}) \cong H^n(\mathbb{C}, \mathbb{Z}(n))$
  • For $F = \mathbb{Q}$, the map $K_3^M(\mathbb{Q}) \to H^3(\mathbb{Q}, \mathbb{Z}(3))$ is not an isomorphism, and the difference is measured by motivic cohomology groups

Importance and research directions

• Étale cohomology groups $H^p(F, \mathbb{Z}(q))$ have a natural product structure
  • Corresponds to the product in Milnor K-theory under the isomorphism $K_n^M(F) \cong H^n(F, \mathbb{Z}(n))$ 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 Friedlander-Suslin spectral sequence and the Levine-Weibel Chow-Witt complex, are also being investigated
• Examples:
  • The Milnor conjecture, which relates Milnor K-theory to quadratic forms and Galois cohomology, was a major open problem that was resolved using ideas from motivic cohomology and the Bloch-Lichtenbaum spectral sequence
  • The Beilinson-Lichtenbaum conjecture, 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