13.2 Motivic cohomology and algebraic K-Theory
5 min read•july 30, 2024
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
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- 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.
Key Terms to Review (23)
Algebraic cycles: Algebraic cycles are formal sums of subvarieties of a given algebraic variety, which play a significant role in the study of both motivic cohomology and algebraic K-Theory. These cycles help in understanding the properties of varieties through their intersection theory and provide a way to relate geometry and topology in a coherent framework. They are essential for defining classes in cohomology theories that correspond to various algebraic invariants.
Algebraic k-theory: Algebraic K-theory is a branch of mathematics that studies projective modules and their relations to algebraic objects through the lens of homotopy theory. It provides tools to analyze algebraic structures like rings and schemes, connecting them with topological concepts, and allows for insights into various mathematical areas such as geometry, number theory, and representation theory.
Beilinson-Soulé Vanishing Conjecture: The Beilinson-Soulé Vanishing Conjecture posits that certain groups of motivic cohomology associated with algebraic varieties vanish in specific degrees, particularly in positive degrees. This conjecture links the realms of motivic cohomology and algebraic K-Theory by suggesting a deeper relationship between algebraic cycles and the structure of these groups. Understanding this conjecture helps illuminate the connections between various cohomology theories and their implications for the study of algebraic geometry.
Bloch-Kato Conjecture: The Bloch-Kato Conjecture is a deep hypothesis in number theory that connects algebraic K-theory and Galois cohomology, suggesting that the Milnor K-theory of a field is related to the Galois cohomology of its fields of fractions. This conjecture has significant implications for understanding the relationship between different types of cohomology theories, particularly in the context of Milnor K-theory, spectral sequences, and arithmetic geometry.
Chow Groups: Chow groups are algebraic structures that capture the idea of algebraic cycles on a variety, facilitating the study of intersection theory and cohomological properties. They provide a way to classify and understand geometric objects in algebraic geometry by associating classes of cycles to each variety, allowing one to analyze relationships between these cycles through operations like addition and intersection. This concept plays a crucial role in linking motivic cohomology and algebraic K-Theory.
Cyclic homology: Cyclic homology is a mathematical concept that arises in the study of noncommutative geometry and algebraic topology, extending the ideas of homology to include cyclic groups. It is primarily used to investigate the structure of algebras and their representations, particularly in relation to algebraic K-theory and motivic cohomology. This theory connects algebraic structures with topological invariants, providing powerful tools for understanding their properties.
Daniel Quillen: Daniel Quillen was a prominent mathematician known for his groundbreaking work in algebraic K-theory, particularly for developing the higher algebraic K-theory framework. His contributions laid the foundation for significant advancements in understanding the relationship between algebraic K-theory and other areas of mathematics, particularly in how these theories intersect with topology, geometry, and arithmetic geometry.
Deligne Cohomology: Deligne cohomology is a mathematical framework that extends traditional cohomology theories by incorporating both topological and algebraic information. It connects algebraic geometry with topology, providing a powerful tool to study the properties of algebraic varieties and their connections to algebraic K-Theory, particularly in the context of motivic cohomology.
Eilenberg-MacLane Spectrum: The Eilenberg-MacLane spectrum is a special kind of spectrum in stable homotopy theory that represents homology theories, specifically $H^n(X; G)$ for an abelian group $G$ and integer $n$. It plays a crucial role in both algebraic K-theory and motivic cohomology by providing a way to study cohomological invariants and their relations to various types of topological spaces.
étale cohomology: Étale cohomology is a powerful tool in algebraic geometry that extends the notion of cohomology to schemes, allowing for the study of algebraic varieties over arbitrary fields. It provides a way to capture topological and algebraic information about these varieties, facilitating connections between geometry and number theory.
Functorial: Functorial refers to the property of a mathematical structure or construction that preserves the relationships between objects in a consistent way when mappings between categories are applied. In the realm of algebraic K-Theory and motivic cohomology, functoriality ensures that various operations and transformations maintain their structure under changes of the underlying objects or spaces.
K-groups: K-groups are algebraic invariants in K-Theory that categorize vector bundles over a topological space or scheme. They provide a way to study and classify these bundles, revealing deep connections between geometry and algebra through various mathematical contexts.
Motivic Cohomology: Motivic cohomology is a mathematical framework that extends classical cohomology theories to the realm of algebraic geometry, providing a way to study algebraic cycles and their properties. It connects various branches of mathematics, including algebraic K-theory and arithmetic geometry, by offering a refined tool for understanding the relationships between different types of geometric objects and their cohomological properties.
Motivic spectral sequence: A motivic spectral sequence is a powerful tool in algebraic geometry and homotopy theory that helps compute motivic cohomology groups through a filtration process. It arises from the study of the relationships between algebraic K-theory and motivic cohomology, providing a systematic way to calculate invariants of algebraic varieties. This tool is particularly valuable in understanding how algebraic structures behave under various morphisms and in different contexts within the motivic framework.
Postnikov Tower: A Postnikov tower is a construction in homotopy theory that breaks down a space into a sequence of stages, each capturing specific homotopical information. It allows mathematicians to study spaces in a systematic way by focusing on their 'homotopy types' at different levels, ultimately connecting to concepts like motivic cohomology and algebraic K-Theory through the lens of stable homotopy theory and spectra.
Regulator Maps: Regulator maps are homomorphisms from algebraic K-theory groups to motivic cohomology groups that help understand the connections between these two important areas in mathematics. These maps serve as a bridge by relating the algebraic properties of schemes to their topological aspects, thus providing a way to transfer information between K-theory and motivic cohomology. This interplay is crucial for understanding how these theories can inform each other, especially in the context of number theory and algebraic geometry.
Smooth schemes: Smooth schemes are algebraic varieties that exhibit nice geometric properties, specifically characterized by having a well-behaved tangent space at every point. This property ensures that the scheme has no 'singularities' or abrupt changes in shape, allowing for a smooth structure that facilitates various mathematical applications, particularly in the realms of K-theory and arithmetic geometry. They play a crucial role in understanding the relationships between algebraic geometry, cohomology theories, and K-theory.
Spectral Sequences: Spectral sequences are powerful computational tools in algebraic topology and homological algebra that allow one to systematically compute the homology or cohomology of complex spaces by breaking them down into simpler pieces. They provide a way to organize and handle information about successive approximations, which can reveal deep relationships between different mathematical structures.
Stable homotopy: Stable homotopy refers to a concept in algebraic topology that studies the properties of spaces and maps that remain invariant under stabilization, typically by adding a dimension. This idea connects to various important results and theories, such as the Thom isomorphism theorem, Bott periodicity, and the relationships between K-theory, bordism, and cobordism theory. It plays a crucial role in understanding algebraic K-theory and its applications to schemes and varieties.
Triangulated categories: Triangulated categories are a type of category in mathematics equipped with an additional structure that allows for the study of homological properties. This structure includes distinguished triangles, which provide a way to relate objects in the category and their morphisms, making them useful in areas like algebraic K-Theory and motivic cohomology.
Vector Bundles: A vector bundle is a topological construction that consists of a base space, typically a manifold, and a vector space attached to every point of the base space, creating a continuous 'family' of vector spaces. This structure allows for a rich interplay between geometry and algebra, enabling concepts like curvature and characteristic classes to be explored through the lens of topology.
Waldhausen's Theorem: Waldhausen's Theorem establishes a connection between the algebraic K-theory of certain types of rings and the structure of their stable homotopy categories. This theorem plays a pivotal role in understanding the K-theory of rings, especially in relation to reduced K-theory and suspension isomorphisms, as well as in the application of motivic cohomology within algebraic K-theory frameworks.
Walter B. V. G. Schmid: Walter B. V. G. Schmid is a mathematician known for his contributions to the fields of algebraic K-theory and motivic cohomology. His work focuses on establishing connections between these two areas, particularly through the lens of homotopy theory and the study of algebraic cycles, which have significant implications in modern algebraic geometry.