14.1 K-Theory of schemes and varieties
3 min read•july 30, 2024
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 , a cutting-edge research area. These groups have a natural filtration called , linked to and the . 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 ()
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
- 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 using the Chern character map, an isomorphism modulo torsion
- K-Theory can be studied using (, ) 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 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
Key Terms to Review (27)
Adams operations: Adams operations are a sequence of operations in K-Theory that act on the K-theory groups of topological spaces, allowing the construction of new classes from existing ones. These operations provide a powerful tool for understanding the structure of K-theory and play a vital role in spectral sequences and periodicity results.
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.
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.
Atiyah-Hirzebruch Spectral Sequence: The Atiyah-Hirzebruch spectral sequence is a powerful tool in algebraic topology that provides a way to compute the K-theory of a space from its cohomology. It connects the geometry of vector bundles to topological invariants, allowing for the classification of vector bundles through the lens of K-theory and characteristic classes.
Brown-Gersten Spectral Sequence: The Brown-Gersten spectral sequence is a powerful computational tool in algebraic K-theory that helps derive K-groups of schemes and varieties from their underlying topological or algebraic structures. It serves as a bridge connecting the algebraic K-theory of a scheme with its étale cohomology, allowing for a deeper understanding of the relationships between these mathematical objects.
Chern character: The Chern character is an important topological invariant associated with complex vector bundles, which provides a connection between K-theory and cohomology. It captures information about the curvature of the vector bundle and its underlying geometric structure, serving as a bridge in various applications, from fixed point theorems to differential geometry.
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 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.
Derived Category: The derived category is a fundamental concept in homological algebra and algebraic geometry that encapsulates information about complexes of objects, such as sheaves or modules, up to quasi-isomorphism. It allows mathematicians to work with chain complexes while focusing on their homological properties, providing a framework to study morphisms and extensions between these complexes. In the context of K-Theory of schemes and varieties, derived categories play a crucial role in understanding the relationships between geometric objects and their associated algebraic structures.
Exact Sequences: Exact sequences are sequences of algebraic objects and morphisms where the image of one morphism equals the kernel of the next. This concept is crucial in understanding how different spaces or structures interact with one another, highlighting relationships such as cohomology and homology. In various contexts, exact sequences can provide powerful tools for studying properties like K-theory and Gysin homomorphisms, as well as their connections to algebraic structures.
Functoriality: Functoriality refers to the principle that relationships between mathematical structures can be preserved through functors, which are mappings between categories that respect the structures involved. This concept is essential in understanding how various K-Theories relate to each other and how different constructions or operations can yield consistent results across different contexts.
Gamma-filtration: Gamma-filtration is a construction in K-theory that provides a way to systematically build a filtration of the K-theory of a ring or a scheme. It is particularly useful for studying the behavior of K-groups under various operations and helps in understanding the connections between algebraic K-theory and other cohomological theories.
Grothendieck's Theorem: Grothendieck's Theorem is a fundamental result in algebraic K-theory, which states that for a smooth projective variety over a field, its K-theory can be expressed in terms of its coherent sheaves. This theorem establishes a deep connection between algebraic geometry and K-theory, leading to insights about the nature of vector bundles and the structure of algebraic varieties.
Homological K-Theory: Homological K-Theory is a branch of algebraic K-theory that focuses on understanding the properties of rings and modules through homological methods. It connects concepts from both algebra and topology, providing insights into how D-branes can be modeled in string theory and how K-theory can classify vector bundles over schemes and varieties.
Isomorphism Theorems: Isomorphism theorems are fundamental results in algebra that describe the structure-preserving relationships between algebraic objects, such as groups, rings, and modules. These theorems provide insights into how different structures can be related to one another through isomorphisms, leading to a deeper understanding of their properties and behaviors. In the context of K-Theory of schemes and varieties, these theorems help establish important connections between different K-groups, facilitating the classification of vector bundles and coherent sheaves.
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.
K-Theory: K-Theory is a branch of mathematics that studies vector bundles and their generalizations through the construction of K-groups, which provide a way to classify and understand vector bundles up to isomorphism. It connects various areas of mathematics, including topology, algebra, and geometry, offering insights into fixed point theorems, quantum field theory, and even string theory.
Lambda-rings: Lambda-rings are algebraic structures that extend the notion of rings by incorporating operations that reflect the behavior of formal power series. They allow for the study of various algebraic invariants, such as K-theory, through a unified framework that accommodates both additive and multiplicative structures.
Michael Atiyah: Michael Atiyah was a prominent British mathematician known for his significant contributions to topology, geometry, and K-Theory. His work laid the groundwork for several important theories and concepts that link abstract mathematics to physical applications, especially in areas like quantum field theory and differential geometry.
Motivic K-Theory: Motivic K-Theory is a branch of algebraic K-theory that studies algebraic varieties and schemes using the tools of homotopy theory and motives. This theory connects classical K-theory with the geometric and topological properties of algebraic objects, allowing for a deeper understanding of their structure and relationships.
Projective Varieties: Projective varieties are a special class of algebraic varieties that can be defined as the zero sets of homogeneous polynomials in a projective space. They are essential in algebraic geometry because they allow for the study of geometric properties through their coordinate systems, which inherently include points at infinity, making them crucial for understanding the compactification of varieties.
Quillen-Lichtenbaum Conjecture: The Quillen-Lichtenbaum Conjecture posits a deep connection between algebraic K-theory and étale cohomology, particularly suggesting that the K-groups of a scheme can be expressed in terms of the étale cohomology groups over that scheme. This conjecture highlights how tools from algebraic topology can provide insights into algebraic geometry and the properties of schemes.
Ring spectrum: A ring spectrum is a type of spectrum in stable homotopy theory that encodes the algebraic structure of a ring while maintaining a topological aspect. It allows for the application of algebraic operations in a homotopical setting, enabling the study of K-theory and related constructs in both stable and unstable environments. This concept is crucial when examining reduced K-theory and the properties of schemes and varieties, as it bridges the gap between topology and algebra.
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.
Topological k-theory: Topological K-theory is a branch of mathematics that studies vector bundles over topological spaces and their classifications, using the language of K-groups. It connects algebraic topology with functional analysis and is pivotal in understanding various phenomena in geometry and topology, linking to concepts like equivariant Bott periodicity and localization theorems, as well as applications in string theory and cobordism.