14.1 K-Theory of schemes and varieties

K-Theory of schemes and varieties is a powerful tool in algebraic geometry. It assigns a graded ring to schemes and varieties, capturing crucial algebraic and geometric information. This approach uses vector bundles or coherent sheaves to define K-Theory groups.

Higher K-Theory groups delve into algebraic K-Theory, a cutting-edge research area. These groups have a natural filtration called gamma-filtration, linked to Adams operations and the Chern character. They're vital for studying algebraic K-Theory and tackling number theory problems.

K-Theory of Schemes and Varieties

Definition and Significance

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• K-Theory assigns a graded ring to a scheme or variety capturing important algebraic and geometric information about the object
• The K-Theory groups are defined using the category of vector bundles or coherent sheaves on the object, with the group operation given by the tensor product
• Provides a powerful tool for studying the geometry and topology of schemes and varieties, as it relates to their algebraic structure and properties
• Closely connected to other important invariants (Chow ring, Picard group) and can be used to compute these invariants in many cases
• Has applications in various areas of algebraic geometry (intersection theory, moduli spaces, algebraic cycles and motives)

Higher K-Theory Groups

• The higher K-Theory groups of a scheme or variety contain information about its algebraic K-Theory, which is a deep and active area of research in modern algebraic geometry
• Have a natural filtration called the gamma-filtration, related to the Adams operations and the Chern character
• Can be used to study the algebraic K-Theory of the object, related to important problems in number theory and arithmetic geometry (Quillen-Lichtenbaum conjecture)

Constructing K-Theory Groups

Definition and Construction

• The K-Theory groups of a scheme or variety X are defined as the Grothendieck groups of the category of vector bundles or coherent sheaves on X, with the group operation given by the tensor product
• The K0 group of X is constructed as the free abelian group generated by isomorphism classes of vector bundles or coherent sheaves on X, modulo the relations given by short exact sequences
• The higher K-Theory groups of X are defined using the Quillen Q-construction or the Waldhausen S-construction, which are categorical constructions that generalize the definition of K0

Computation Techniques

• The K-Theory groups can be computed using the Grothendieck-Riemann-Roch theorem, which relates the Chern character of a vector bundle to its class in K-Theory
• For smooth varieties over a field, the K-Theory groups can be related to the Chow groups using the Chern character map, an isomorphism modulo torsion
• K-Theory can be studied using spectral sequences (Atiyah-Hirzebruch spectral sequence, Brown-Gersten spectral sequence) which relate the K-Theory to other cohomology theories

Properties of K-Theory Groups

Algebraic Structure

• The K-Theory groups have a rich algebraic structure, including a graded ring structure given by the tensor product and exterior power operations
• The K-Theory of a smooth variety over a field is a contravariant functor with respect to morphisms of varieties and satisfies certain functorial properties (projection formula, homotopy invariance)
• The K-Theory of a regular scheme is closely related to its Picard group and its Chow ring, with natural maps between these objects that are isomorphisms in certain cases

Duality and Lambda-Rings

• The K-Theory of a scheme or variety satisfies certain duality theorems (Poincaré duality for smooth varieties over a field) which relates the K-Theory to the K-Theory with compact supports
• K-Theory can be studied using the theory of lambda-rings and the Adams operations, providing a powerful tool for understanding the structure of the K-Theory groups

Applications of K-Theory

Geometric Applications

• Proves the Grothendieck-Riemann-Roch theorem for schemes and varieties, relating the Chern character of a vector bundle to its class in K-Theory and providing a tool for computing intersection numbers
• Studies the geometry of algebraic cycles and motives, proving results about the Chow groups and the Picard group
• Applies to the study of moduli spaces of vector bundles and coherent sheaves on a variety, constructing invariants of these moduli spaces (Donaldson invariants)

Topological and Birational Applications

• Proves results about the topology of schemes and varieties (Kodaira vanishing theorem, Lefschetz hyperplane theorem) by studying the behavior of vector bundles and coherent sheaves under certain operations
• Provides invariants that are preserved under birational equivalence and can be used to measure the complexity of the object, applied to the study of birational geometry and the minimal model program