13.2 Motivic cohomology and algebraic K-Theory

Motivic cohomology and algebraic K-theory are powerful tools for studying algebraic varieties. They provide insights into the intrinsic properties of these structures, going beyond what other cohomology theories can capture. The motivic spectral sequence links these two theories.

This connection allows us to compute algebraic K-theory groups using motivic cohomology. It's a key technique in modern algebraic geometry, helping us understand vector bundles, algebraic cycles, and other important structures on varieties.

Motivic cohomology and K-theory

Definition and properties of motivic cohomology

Top images from around the web for Definition and properties of motivic cohomology

• 

  L∞-algebras and their cohomology | Emergent Scientist View original

  Is this image relevant?

• 

  L∞-algebras and their cohomology | Emergent Scientist View original

  Is this image relevant?

• 

  L∞-algebras and their cohomology | Emergent Scientist View original

  Is this image relevant?

• 

  L∞-algebras and their cohomology | Emergent Scientist View original

  Is this image relevant?

• 

  L∞-algebras and their cohomology | Emergent Scientist View original

  Is this image relevant?

1 of 3

Top images from around the web for Definition and properties of motivic cohomology

• 

  L∞-algebras and their cohomology | Emergent Scientist View original

  Is this image relevant?

• 

  L∞-algebras and their cohomology | Emergent Scientist View original

  Is this image relevant?

• 

  L∞-algebras and their cohomology | Emergent Scientist View original

  Is this image relevant?

• 

  L∞-algebras and their cohomology | Emergent Scientist View original

  Is this image relevant?

• 

  L∞-algebras and their cohomology | Emergent Scientist View original

  Is this image relevant?

1 of 3

• Motivic cohomology is a cohomology theory for algebraic varieties that takes into account the underlying algebraic structure and geometry of the variety
• Defined using algebraic cycles and can be thought of as a generalization of Chow groups
  • Chow groups are a way to measure the size of algebraic cycles on a variety
  • Motivic cohomology extends this idea to include more general algebraic structures
• Provides a way to study the intrinsic properties of algebraic varieties that are not captured by other cohomology theories (étale cohomology, singular cohomology)
• Motivic cohomology groups are contravariant functors from the category of smooth schemes over a field to the category of abelian groups

Connection to algebraic K-theory

• Close relationship between motivic cohomology and algebraic K-theory
• Motivic cohomology provides a way to compute algebraic K-theory groups
  • Algebraic K-theory groups measure the structure of vector bundles and other algebraic objects on a variety
  • Motivic cohomology can be used to construct explicit classes in algebraic K-theory
• Connection is established through the motivic spectral sequence, which relates the two theories
  • Spectral sequence is a tool for computing homology or cohomology groups by breaking them down into simpler pieces
  • Motivic spectral sequence starts with motivic cohomology and converges to algebraic K-theory
• Motivic cohomology can be used to define regulator maps from algebraic K-theory to other cohomology theories (Deligne cohomology, étale cohomology)
  • Regulator maps provide a way to compare different cohomology theories and understand their relationships
  • Example: Beilinson regulator map relates algebraic K-theory to Deligne cohomology

Motivic spectral sequence

Construction of the motivic spectral sequence

• Spectral sequence that relates motivic cohomology to algebraic K-theory
• Constructed using the motivic Eilenberg-MacLane spectrum and the motivic Postnikov tower
  • Eilenberg-MacLane spectrum is a spectrum that represents a cohomology theory
  • Postnikov tower is a way to approximate a spectrum by a sequence of simpler spectra
• Convergent spectral sequence, meaning that it converges to the algebraic K-theory groups of the variety under consideration
• E2-page of the spectral sequence is given by the motivic cohomology groups of the variety
  • E2-page is the second page of the spectral sequence and contains important information about the cohomology groups being computed
• Abutment of the spectral sequence is the algebraic K-theory groups

Properties and applications

• Differentials in the motivic spectral sequence provide information about the relationship between motivic cohomology and algebraic K-theory
  • Differentials are maps between the different pages of the spectral sequence that encode important structural information
  • Can be used to compute algebraic K-theory groups from motivic cohomology
• Functorial with respect to morphisms of varieties
  • Allows for the study of the behavior of algebraic K-theory under morphisms (pullbacks, pushforwards)
  • Important for understanding the functorial properties of algebraic K-theory
• Can be used to prove vanishing theorems for algebraic K-theory
  • Example: Beilinson-Soulé vanishing conjecture predicts that certain motivic cohomology groups vanish, which implies vanishing results for algebraic K-theory
• Provides a powerful tool for studying the relationship between algebraic cycles and vector bundles on a variety

K-theory computation

Computing algebraic K-theory using the motivic spectral sequence

• Motivic spectral sequence provides a powerful tool for computing algebraic K-theory groups of algebraic varieties
• Process involves several steps:
  1. Compute the motivic cohomology groups of the variety, which form the E2-page of the spectral sequence
  2. Determine the differentials in the spectral sequence, which provide information about the relationship between motivic cohomology and algebraic K-theory
  3. Use the differentials to compute the algebraic K-theory groups, which appear as the abutment of the spectral sequence
• In some cases, the motivic spectral sequence degenerates, meaning that all differentials vanish
  • In this case, the algebraic K-theory groups can be read off directly from the E2-page
  • Degeneration is a particularly nice situation that simplifies the computation of algebraic K-theory

Examples and applications

• Computation of algebraic K-theory using the motivic spectral sequence often involves the use of additional tools (Bloch-Kato conjecture, Beilinson-Soulé vanishing conjecture)
  • These conjectures provide additional information about the structure of motivic cohomology and algebraic K-theory that can be used in computations
• Examples of varieties for which the algebraic K-theory has been computed using the motivic spectral sequence:
  • Smooth projective varieties over fields
  • Certain singular varieties (nodal cubic curve)
• Applications of algebraic K-theory computations:
  • Study of vector bundles and algebraic cycles on varieties
  • Computation of obstruction groups for the existence of certain types of algebraic structures (division algebras, quadratic forms)
  • Investigation of the relationship between algebraic K-theory and other invariants of varieties (Chow groups, Hodge theory)

Motivic cohomology vs other theories

Relationship to other cohomology theories

• Motivic cohomology is related to several other important cohomology theories in algebraic geometry
  • Chow groups
  • Étale cohomology
  • Deligne cohomology
• Regulator maps from motivic cohomology to these other cohomology theories
  • Provide a way to compare the different theories and understand their relationships
  • Example: regulator map from motivic cohomology to Deligne cohomology relates algebraic K-theory to Deligne cohomology
• Beilinson-Soulé vanishing conjecture
  • Predicts that the regulator map from motivic cohomology to Deligne cohomology is an isomorphism in certain degrees
  • Has important consequences for the computation of algebraic K-theory
• Motivic cohomology can also be related to other cohomology theories through the use of spectral sequences (Atiyah-Hirzebruch spectral sequence, Bloch-Ogus spectral sequence)

Current research and open problems

• Understanding the relationships between motivic cohomology and other cohomology theories is an active area of research in algebraic geometry
• Has important applications to problems such as:
  • Bloch-Kato conjecture, which relates motivic cohomology to étale cohomology and Milnor K-theory
  • Milnor conjecture, which relates quadratic forms to étale cohomology and motivic cohomology
• Open problems and conjectures:
  • Suslin's conjecture, which predicts that certain motivic cohomology groups are isomorphic to étale cohomology groups
  • Voevodsky's conjecture, which predicts that the motivic Steenrod algebra is isomorphic to the classical Steenrod algebra
  • Friedlander-Mazur conjecture, which predicts that the motivic homotopy groups of the sphere spectrum are isomorphic to the classical stable homotopy groups of spheres
• Resolving these conjectures and understanding the precise relationships between motivic cohomology and other theories is a major goal of current research in this area.