13.3 Applications to algebraic cycles and motivic cohomology
4 min read•july 30, 2024
Higher algebraic K-theory provides powerful tools for studying algebraic cycles and . These concepts are crucial in algebraic geometry, offering insights into the structure of varieties and their geometric properties.
This section explores how K-theory connects to algebraic cycles and Chow groups. It also delves into motivic cohomology, a theory that combines algebraic and geometric information, and its relationship to other important invariants in algebraic geometry.
Algebraic K-theory for Cycles and Chow Groups
Applying Algebraic K-theory to Algebraic Cycles
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- Algebraic K-theory provides a powerful framework for studying algebraic cycles and Chow groups, which are central objects in algebraic geometry
- The higher K-groups, particularly the Milnor K-theory and Quillen K-theory, capture important information about algebraic cycles and their equivalence relations (rational equivalence, algebraic equivalence)
- The Chow ring of a smooth projective variety can be expressed in terms of the Milnor K-theory of its function field, establishing a direct connection between K-theory and algebraic cycles
- The Bloch-Quillen formula relates the of codimension p cycles on a smooth variety to the p-th Quillen K-group of its coordinate ring, providing a bridge between algebraic cycles and K-theory
Grothendieck-Riemann-Roch Theorem
- The Riemann-Roch theorem for higher K-theory, known as the Grothendieck-Riemann-Roch theorem, relates the Chow ring and K-theory, allowing for the computation of characteristic classes of algebraic cycles
- The theorem expresses the Chern character of a vector bundle in terms of its algebraic K-theory class and provides a powerful tool for studying the intersection theory of algebraic cycles
- Applications of the Grothendieck-Riemann-Roch theorem include the computation of of vector bundles, the study of the Hodge conjecture, and the proof of the Adams-Riemann-Roch theorem in arithmetic geometry
Motivic Cohomology and Algebraic Cycles
Defining Motivic Cohomology
- Motivic cohomology is a cohomology theory for algebraic varieties that incorporates both algebraic and geometric information, making it a natural tool for studying algebraic cycles
- The motivic cohomology groups of a variety can be defined using the higher Chow groups, which are a generalization of the classical Chow groups and capture information about algebraic cycles
- The motivic cohomology of a variety satisfies a number of important properties, such as homotopy invariance, Mayer-Vietoris sequence, and Gysin long exact sequence, which are crucial for studying the behavior of algebraic cycles under various geometric constructions (blowups, fibrations)
Relating Motivic Cohomology to Other Invariants
- The motivic cohomology groups of a variety can be related to its algebraic K-theory groups through the motivic spectral sequence, providing a powerful tool for computing invariants of algebraic cycles
- The conjectures of Beilinson and Lichtenbaum relate the motivic cohomology of a variety to its and provide a deep connection between algebraic cycles and arithmetic geometry
- The motivic cohomology of a variety is closely related to its Hodge theory, with the motivic Hodge conjecture providing a framework for studying the Hodge structures of algebraic cycles
Motivic Spectral Sequence for Cohomology
Construction and Properties
- The motivic spectral sequence is a powerful tool for computing the motivic cohomology groups of an algebraic variety, relating them to more accessible invariants such as algebraic K-theory and étale cohomology
- The motivic spectral sequence arises from the slice filtration on the motivic , which allows for a systematic study of the layers of motivic cohomology
- The E2-page of the motivic spectral sequence is given by the motivic cohomology groups of the variety, while the E∞-page is related to the algebraic K-theory groups, providing a means to compute motivic cohomology from K-theory
Applications and Computations
- The differentials in the motivic spectral sequence encode important information about the relationship between algebraic cycles and K-theory, and their computation often requires deep results from algebraic geometry and homotopy theory (Voevodsky's proof of the Milnor conjecture)
- The motivic spectral sequence has been used to prove important results in the theory of algebraic cycles, such as the Bloch-Kato conjecture and the Beilinson-Lichtenbaum conjecture, highlighting its significance in the field
- Explicit computations of the motivic spectral sequence have been carried out for certain classes of varieties, such as curves, surfaces, and toric varieties, providing insights into the structure of their motivic cohomology groups
Motivic Cohomology in Algebraic Varieties
Invariants and Structures
- Motivic cohomology provides a powerful set of tools for studying the geometry and arithmetic of algebraic varieties, with applications ranging from the classification of varieties to the study of rational points
- The motivic cohomology groups of a variety can be used to define important invariants, such as the motivic zeta function and the motivic Euler characteristic, which capture information about the geometry and singularities of the variety (Hodge-Deligne polynomial)
- The motivic cohomology of a variety is closely related to its Hodge theory, with the motivic Hodge conjecture providing a deep connection between the two theories and a framework for studying the Hodge structures of algebraic cycles
Applications to Geometry and Arithmetic
- Motivic cohomology has been used to study the rationality and stable rationality of algebraic varieties, with the motivic obstruction to rationality providing a powerful tool for detecting non-rational varieties (Artin-Mumford example)
- The theory of motivic integration, which extends classical p-adic integration to the motivic setting, has found important applications in the study of birational geometry and the minimal model program, highlighting the significance of motivic cohomology in modern algebraic geometry
- Motivic cohomology has also been applied to the study of rational points on algebraic varieties, with the Brauer-Manin obstruction and the descent obstruction providing effective methods for determining the existence and density of rational points (Châtelet surfaces)
Key Terms to Review (14)
Alexander Grothendieck: Alexander Grothendieck was a revolutionary French mathematician known for his significant contributions to algebraic geometry, homological algebra, and K-theory. His work fundamentally shaped modern mathematics, particularly through the development of the Grothendieck group and the insights into K-theory that link algebraic structures with topological concepts.
Chern classes: Chern classes are a set of characteristic classes associated with complex vector bundles, providing vital topological invariants that help classify vector bundles over a manifold. They connect deeply with various fields such as geometry, topology, and algebraic geometry, allowing us to analyze vector bundles through their topological properties.
Chow Group: Chow groups are mathematical structures that capture the idea of algebraic cycles on a projective variety, allowing for the classification and study of these cycles in a rigorous way. They provide a way to connect geometry with algebraic topology and help in understanding properties of varieties, especially in the context of intersection theory and motives.
Cycle class map: The cycle class map is a homomorphism that connects algebraic cycles on a variety to cohomology classes, typically in the context of Chow groups. This map plays a crucial role in bridging the world of algebraic geometry with topology, as it allows one to translate geometric properties into algebraic invariants. By linking cycles to their corresponding classes in motivic cohomology, the cycle class map also facilitates deeper investigations into the relationship between cycles and other mathematical structures.
Derived categories: Derived categories are a fundamental concept in homological algebra and algebraic geometry, providing a way to systematically study complexes of objects and their morphisms. They allow mathematicians to manage and understand derived functors, cohomology theories, and triangulated structures, enabling deeper insights into algebraic cycles and motivic cohomology as well as applications in arithmetic geometry.
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.
Grothendieck's Riemann-Roch Theorem: Grothendieck's Riemann-Roch Theorem is a fundamental result in algebraic geometry that extends classical results about Riemann surfaces to higher-dimensional varieties. It provides a powerful way to calculate the dimension of certain spaces of sections of line bundles and offers deep insights into the intersection theory of algebraic cycles. This theorem connects various areas of mathematics, including topology, algebraic cycles, and arithmetic geometry, demonstrating relationships between geometric properties and cohomological data.
K-cycle: A k-cycle is a specific type of algebraic cycle that represents a class of geometric objects associated with a smooth projective variety over a field. These cycles are crucial in studying the properties of algebraic varieties and their relationships to cohomological theories, including motivic cohomology.
K-theory of schemes: K-theory of schemes is a branch of mathematics that studies algebraic varieties through the lens of vector bundles and projective modules, providing tools to classify and understand these structures. It connects with various important concepts such as stable isomorphism classes of vector bundles and the behavior of coherent sheaves, paving the way for deeper insights in algebraic cycles and motivic cohomology.
Lichtenstein's Theorem: Lichtenstein's Theorem is a result in algebraic geometry that provides a relationship between algebraic cycles and motivic cohomology. It essentially states that under certain conditions, the numerical equivalence of algebraic cycles corresponds to the vanishing of a certain class in motivic cohomology. This connection is significant as it links different mathematical frameworks, offering insights into the structure and properties of algebraic cycles.
Motivic Class: A motivic class is an equivalence class of algebraic cycles, used in the context of algebraic geometry and motivic cohomology to study the relationships between different cycles on a variety. It helps organize these cycles into a structured framework where they can be analyzed using tools from both algebraic and topological perspectives, leading to insights into their geometric properties and connections to cohomology theories.
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.
Stable homotopy category: The stable homotopy category is a mathematical framework that generalizes the concept of homotopy theory by focusing on stable phenomena, particularly in the context of spectra. In this category, morphisms are defined up to stable homotopy, allowing for the study of objects like vector bundles and algebraic cycles through a lens that considers their behavior under stabilization. This perspective is crucial for understanding deeper relationships between topology, geometry, and algebra, particularly when discussing concepts like motivic cohomology and its applications.
Vladimir Voevodsky: Vladimir Voevodsky was a prominent Russian mathematician known for his groundbreaking work in algebraic geometry, homotopy theory, and K-theory. His innovative ideas and methods transformed the landscape of mathematics, particularly in the areas of algebraic cycles and motivic cohomology, where he contributed significantly to the understanding of how these concepts relate to various cohomology theories.