and 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 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 , , 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 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 (, singular cohomology)
Motivic cohomology groups are contravariant functors from the category of 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 from algebraic K-theory to other cohomology theories (, 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 and the motivic
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
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: 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:
Compute the motivic cohomology groups of the variety, which form the E2-page of the spectral sequence
Determine the differentials in the spectral sequence, which provide information about the relationship between motivic cohomology and algebraic K-theory
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 (, 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 (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 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.