6.2 The Mayer-Vietoris sequence in K-theory
3 min read•august 16, 2024
The is a powerful tool in K-theory, linking of a scheme to those of its open subschemes. It forms a , allowing us to break down complex schemes into simpler parts for easier computation.
This sequence is crucial for calculating K-groups of intricate schemes, especially those built by together simpler pieces. It's a special case of the and extends to other theories, making it a versatile tool in algebraic geometry.
Mayer-Vietoris Sequence for K-theory
Definition and Structure
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- Mayer-Vietoris sequence relates K-groups of a scheme to K-groups of its open subschemes
- For scheme X with open subschemes U and V where X = U ∪ V, sequence forms long exact sequence of K-groups
- Sequence takes form
- Maps in sequence induced by inclusion and restriction of
- Sequence extends infinitely in both directions, connecting all higher and lower K-groups
Key Properties and Applications
- Generalizes to other cohomology theories (singular cohomology, étale cohomology)
- Special case of localization sequence in K-theory
- Fundamental tool for computing K-groups of complex schemes
- Requires understanding of exact sequences and K-groups for schemes
- Allows decomposition of schemes into simpler open subschemes for easier computation
- Useful for schemes constructed by gluing together or well-understood building blocks (, )
Exactness of Mayer-Vietoris Sequence
Proof Techniques
- Proof relies on constructing K-theory as homotopy groups of spectrum associated with scheme
- Key step involves showing sequence of forms homotopy fibration
- Utilizes and properties of exact categories of vector bundles on schemes
- Demonstrates at each stage by analyzing kernels and images of maps in sequence
- Shows elements in kernel of one map belong to image of previous map
- Employs techniques (diagram chasing, snake lemma)
Implications and Extensions
- Proof establishes Mayer-Vietoris sequence as natural with respect to morphisms of schemes
- Exactness ensures reliability of sequence for K-group computations
- Extends to other cohomology theories with similar Mayer-Vietoris sequences (, )
- Provides foundation for more advanced K-theory constructions (localization sequence, Gersten resolution)
Computing K-groups with Mayer-Vietoris
Computational Strategies
- Decompose scheme into simpler open subschemes with known or easily computable K-groups
- Use long exact sequence structure to determine unknown K-groups from known subscheme K-groups
- Leverage (K_{n-1} term) for recursive K-group computations
- Carefully track maps, kernels, and images to determine structure of unknown K-groups
- Apply sequence iteratively for complex schemes requiring multiple decompositions
Examples and Applications
- Compute K-groups of projective spaces (P^n) using affine cover
- Determine K-theory of (, )
- Calculate K-groups of toric varieties using combinatorial data
- Analyze K-theory of schemes obtained by gluing affine pieces ()
- Investigate K-groups of through and Mayer-Vietoris
Mayer-Vietoris vs Other K-theory Tools
Connections to Other Sequences
- Closely related to localization sequence dealing with closed subschemes and complements
- Forms special case of descent spectral sequence for K-theory in certain situations
- Analogous to Mayer-Vietoris sequences in other cohomology theories (singular cohomology, étale cohomology)
- Connects to spectral sequences used in ()
Advanced Applications
- Plays role in relating algebraic K-theory to topological K-theory for complex varieties
- Fundamental in developing advanced K-theory concepts (, )
- Aids in choosing appropriate technique for specific K-theory computations
- Facilitates comparison between algebraic and topological K-theory (, )
- Enables study of K-theory for more general geometric objects (stacks, derived schemes)
Key Terms to Review (31)
Affine schemes: Affine schemes are the basic building blocks in the study of algebraic geometry, representing the spectrum of a commutative ring. They provide a way to link algebraic properties with geometric structures, allowing for a deep understanding of how algebraic equations translate into geometric shapes. In the context of K-theory, affine schemes serve as crucial examples that help illustrate important concepts, such as local properties and cohomology.
Algebraic Varieties: Algebraic varieties are geometric objects defined as the solutions to polynomial equations over a field, often studied in algebraic geometry. They play a crucial role in connecting algebraic structures with geometric interpretations, which helps in understanding many aspects of mathematics including number theory and topology.
Atiyah-Hirzebruch spectral sequence: The Atiyah-Hirzebruch spectral sequence is a powerful tool in algebraic topology and algebraic K-theory that provides a way to compute the K-groups of a space by relating them to the homology of that space. This sequence connects various mathematical concepts, allowing for deeper insights and computations, particularly in the study of vector bundles and characteristic classes.
Bloch-Lichtenbaum Spectral Sequence: The Bloch-Lichtenbaum spectral sequence is a mathematical tool used in algebraic K-theory, particularly to compute K-groups from a filtered object, such as a presheaf of abelian groups. It arises in the study of motives and relates different levels of K-theory through a spectral sequence that converges to the K-theory groups of a scheme or a space, revealing connections between local and global properties.
Bott periodicity: Bott periodicity is a fundamental phenomenon in stable homotopy theory and algebraic K-theory, stating that the K-theory of complex vector bundles exhibits periodic behavior with a period of 2. This means that when one studies the K-theory of spheres, particularly complex projective spaces, one finds that the results repeat every two dimensions, leading to powerful simplifications in calculations and applications across various fields.
Čech Cohomology: Čech cohomology is a tool in algebraic topology that studies the global properties of a topological space by examining the behavior of continuous functions defined on open covers of that space. It provides a way to compute cohomology groups, which are algebraic structures capturing information about the shape and features of the space, and can be used to relate topological properties to algebraic constructs. In the context of K-theory, Čech cohomology plays a crucial role in establishing relationships between different spaces through spectral sequences and exact sequences, such as the Mayer-Vietoris sequence.
Chern Character: The Chern character is a topological invariant associated with vector bundles that encodes information about their curvature and cohomology classes. It acts as a bridge between K-theory and cohomology, facilitating the computation of topological invariants and relationships in various mathematical contexts.
Cohomology: Cohomology is a mathematical concept used in algebraic topology and algebraic K-theory to study the properties of topological spaces through the use of cochains and cohomology groups. It provides a way to associate algebraic invariants to topological spaces, which can help understand their structure and relationships. Cohomology plays a crucial role in various frameworks, allowing for the computation of K-groups and the application of spectral sequences, among other uses.
De Rham Cohomology: De Rham cohomology is a mathematical tool used to study the topology of smooth manifolds by associating differential forms with cohomology groups. It provides a bridge between differential geometry and algebraic topology, allowing us to analyze the properties of manifolds through the behavior of these forms. The main idea is that closed forms represent classes in cohomology, enabling deeper insights into the structure and characteristics of manifolds.
Del Pezzo surfaces: Del Pezzo surfaces are a specific class of algebraic surfaces that are characterized by having ample anticanonical bundles. These surfaces play a significant role in algebraic geometry and can be classified based on their degree, which corresponds to the number of lines on the surface. Their rich structure and properties make them important in various applications, including K-theory and the study of rational points.
Dimensional Shifts: Dimensional shifts refer to the phenomenon where the dimensionality of algebraic objects changes as one moves through a topological space or as one applies various operations in K-theory. This concept is essential for understanding how certain algebraic invariants behave when considering different open coverings and their intersections in a space, particularly in the context of the Mayer-Vietoris sequence.
Elliptic Curves: Elliptic curves are smooth, projective algebraic curves of genus one, equipped with a specified point, typically defined over a field. They play a crucial role in various areas of mathematics, including number theory, cryptography, and algebraic geometry, acting as a bridge between these disciplines and allowing for deep connections to other mathematical concepts.
Exactness: Exactness refers to a property of sequences in mathematics, particularly in the context of homological algebra, where a sequence is said to be exact if the image of one morphism equals the kernel of the next. This concept plays a crucial role in understanding how mathematical structures relate to each other and is key to working with resolutions and derived functors.
Gluing: In the context of K-theory, gluing refers to the process of combining algebraic structures or topological spaces by specifying how they connect along shared boundaries. This concept is crucial in understanding how different pieces of spaces or objects can be put together to form a larger, coherent entity, allowing for the application of various tools in algebraic K-theory, particularly when using the Mayer-Vietoris sequence.
Hirzebruch surfaces: Hirzebruch surfaces are a class of algebraic surfaces that can be constructed as $ ext{P}^1$-bundles over the projective line $ ext{P}^1$. They are important in algebraic geometry and K-theory due to their simple structure and the way they allow for the study of various geometric properties. These surfaces have applications in understanding the topology of algebraic varieties and play a role in constructing more complex surfaces through their properties.
Homological Algebra: Homological algebra is a branch of mathematics that studies homology in a general algebraic setting, focusing on the relationships and structures between algebraic objects through sequences and functors. This area connects various aspects of mathematics, allowing for the investigation of properties such as exactness, which are essential in understanding complex algebraic systems and their interrelations.
Homotopy Type: Homotopy type refers to the classification of topological spaces based on their homotopy equivalence, which means that two spaces can be continuously deformed into each other. This concept is crucial in algebraic topology as it allows mathematicians to study spaces using more manageable algebraic structures, like groups or rings, by focusing on their intrinsic properties rather than their specific geometric configurations. Understanding homotopy types connects to key concepts in various areas, including K-theory, where one investigates the relationships between different algebraic structures and their associated topological properties.
K-groups: K-groups are algebraic constructs in K-theory that classify vector bundles over a topological space or schemes in algebraic geometry. These groups provide a way to study the structure of these objects and their relationships to other mathematical concepts, connecting various areas of mathematics including topology, algebra, and number theory.
K-theory computations: K-theory computations refer to the calculations and techniques used to determine the K-theory groups of algebraic or topological spaces, providing a bridge between geometry and algebra. These computations often leverage powerful tools such as exact sequences, spectral sequences, and localization techniques, which help in understanding the structure of K-groups. The results of these computations have implications in various fields, particularly in connecting homological algebra with stable homotopy theory.
Localization sequence: The localization sequence is a critical tool in algebraic K-theory that captures the relationship between K-theory groups of a space and its localization with respect to a certain set of morphisms. This sequence is especially significant as it illustrates how one can connect global properties of a space with local behavior, ultimately leading to deeper insights into the structure of K-groups and their computations.
Long Exact Sequence: A long exact sequence is a sequence of abelian groups and homomorphisms that represents the relationships between the groups involved in a short exact sequence, extending infinitely in both directions while preserving the property of exactness. This concept is crucial in understanding how various algebraic structures connect with one another, especially in contexts involving resolutions, abelian categories, and specific applications like K-theory.
Mayer-Vietoris Sequence: The Mayer-Vietoris sequence is a powerful tool in algebraic topology that helps in computing the homology and K-theory of a space by breaking it down into simpler pieces. It provides a way to relate the K-groups of a space to those of its open covers, which is crucial for understanding properties of complex spaces and their decompositions.
Motivic cohomology: Motivic cohomology is a homological invariant in algebraic geometry that connects the geometry of algebraic varieties to algebraic K-theory and Galois cohomology. It generalizes classical cohomological theories and provides a framework for understanding relationships between different areas of mathematics, including topology and number theory.
Projective spaces: Projective spaces are mathematical structures that arise in the study of geometry, algebra, and topology, formed by taking a vector space and adding 'points at infinity' to it. This concept allows for the exploration of properties that remain invariant under projection, linking various mathematical disciplines and providing a framework for understanding higher-dimensional geometric objects.
Quillen's Q-construction: Quillen's Q-construction is a method for associating a spectrum to a given category, especially in the context of algebraic K-theory. It serves as a bridge between homotopy theory and algebraic structures, allowing for the computation of K-theory by utilizing simplicial sets and their associated topological spaces. This construction plays a crucial role in understanding the relationship between different topological and algebraic invariants.
Resolution of singularities: Resolution of singularities is a process in algebraic geometry that transforms a variety with singular points into a new variety that is smooth (non-singular) in a way that retains the essential structure of the original. This technique is crucial for understanding the behavior of algebraic varieties and plays a significant role in various applications, including K-theory and topology.
Singular varieties: Singular varieties are algebraic varieties that contain points where the local structure is not well-defined, often characterized by points where the variety fails to be smooth. These singularities can complicate the understanding of geometric and topological properties and have important implications in K-theory, particularly in calculating K-groups and studying the behavior of sheaves on such varieties.
Spectra: In algebraic K-theory, spectra are structured objects that generalize the notion of spaces or sets, allowing us to study stable homotopy theory and related constructions. They serve as a bridge between algebraic and topological methods, providing a framework for understanding K-theory via stable categories and homotopy types. Spectra enable us to encode information about vector bundles, their intersections, and transformations in a coherent manner.
Topological Spaces: Topological spaces are mathematical structures that allow for the formal study of concepts such as continuity, convergence, and compactness. They consist of a set of points along with a collection of open sets that satisfy specific axioms, enabling a general framework for discussing geometric and spatial properties. This concept is crucial for understanding various topics in algebraic K-theory, particularly in the construction of the Grothendieck group K0 and in the application of the Mayer-Vietoris sequence, which both rely on the properties of spaces defined within this framework.
Toric varieties: Toric varieties are a special class of algebraic varieties that are defined by combinatorial data, particularly through fans or polyhedral cones. They arise from the theory of torus actions on varieties, allowing for a geometric representation of combinatorial structures and providing powerful tools for calculations in K-theory.
Vector Bundles: Vector bundles are mathematical structures that consist of a family of vector spaces parameterized by a topological space. They play a crucial role in connecting algebraic topology, differential geometry, and algebraic K-theory, serving as a way to study vector fields and their properties over various spaces.