13.3 Applications to algebraic cycles and motivic cohomology

Higher algebraic K-theory provides powerful tools for studying algebraic cycles and motivic cohomology. 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 Chow group 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 Chern classes 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 étale cohomology 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 stable homotopy category, 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)