10.2 The Quillen-Lichtenbaum conjecture
4 min read•august 16, 2024
The links and for . It suggests that for over , the for K-theory becomes an isomorphism after tensoring with Zp for large primes p.
This conjecture is a key part of understanding the relationship between K-theory and . It offers a powerful method for computing higher algebraic using étale cohomology, potentially providing new insights in both algebraic geometry and .
Quillen-Lichtenbaum Conjecture
Formulation and Relation to Étale Cohomology
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- Quillen-Lichtenbaum conjecture postulates a deep connection between algebraic K-theory and étale cohomology for arithmetic schemes
- Conjecture states for a smooth variety X over a finite field, the étale descent for K-theory becomes an isomorphism after tensoring with Zp for sufficiently large primes p
- Étale cohomology generalizes and plays a crucial role in the conjecture's formulation
- Involves comparison between of algebraic K-groups and certain étale cohomology groups
- Relationship between K-theory and étale cohomology expressed through a spectral sequence (key tool in studying the conjecture)
- Understanding the conjecture requires familiarity with and étale cohomology theory
- Higher algebraic K-theory concepts (K-groups, )
- Étale cohomology fundamentals (étale topology, cohomology groups)
Mathematical Framework and Components
- Conjecture utilizes p-adic completion of K-groups
- Process involves tensoring K-groups with
- Allows focus on p-primary torsion information
- Étale cohomology groups involved typically have
- Often use or as coefficient rings
- Spectral sequence central to the conjecture
- Convergence of this sequence key to the conjecture's statement
- Smooth varieties over finite fields serve as primary geometric objects
- Examples (, , )
Implications for K-Groups
Computational Advantages
- Provides powerful method for computing higher algebraic K-groups using étale cohomology (often more tractable)
- Suggests p-adic K-theory of arithmetic schemes can be effectively computed using étale cohomology techniques
- Implies certain K-groups can be understood in terms of Galois cohomology (connection to number theory)
- Verification would allow more efficient calculations of K-groups, especially in higher degrees
- Example K-groups ()
- Establishes strong link between algebraic geometry and arithmetic (potential new insights in both fields)
Structural Insights
- Conjecture has implications for understanding the structure of algebraic K-groups
- Particularly illuminates torsion subgroups
- Provides framework for analyzing p-adic behavior of K-groups
- Example p-adic phenomena (, )
- Suggests deep connections between K-theory and arithmetic geometry
- Links to special values of
- Connections to and
Quillen-Lichtenbaum vs Serre Conjectures
Focus and Scope
- Quillen-Lichtenbaum conjecture focuses on relationship between K-theory and étale cohomology
- Serre conjecture deals with Galois representations
- Quillen-Lichtenbaum concerns
- Serre conjecture addresses modularity of odd 2-dimensional Galois representations
Impact and Status
- Both conjectures have significantly impacted their respective fields
- Motivated substantial research efforts in algebraic geometry and number theory
- Serre conjecture proven (part of Serre's modularity conjecture)
- Quillen-Lichtenbaum conjecture remains open in full generality
- Partial results known for specific cases (finite fields, certain number fields)
Methodological Differences
- Quillen-Lichtenbaum conjecture requires advanced techniques from algebraic K-theory
- Spectral sequences, motivic cohomology, homotopy theory
- Serre conjecture primarily uses methods from number theory and representation theory
- Modular forms, Galois representations, automorphic forms
- Both involve deep connections between different areas of mathematics
- Quillen-Lichtenbaum (K-theory, cohomology, arithmetic geometry)
- Serre (number theory, representation theory, algebraic geometry)
Proof Strategies for the Conjecture
Motivic Cohomology Approach
- Utilizes motivic cohomology and its relationship to both K-theory and étale cohomology
- (now a theorem) plays crucial role in many approaches
- Connects to Galois cohomology
- provides framework for relating different cohomology theories
Homotopy Theory Techniques
- Employs spectrum-level constructions in attempts to prove the conjecture
- Utilizes and ∞-categorical methods
- Investigates of K-theory spectra
- Example Construction of étale K-theory spectrum
Field-Specific Approaches
- Study of conjecture for specific types of fields yields partial results and insights
- Number fields ( with characteristic 0)
- Finite fields (fields with prime power order)
- Proofs involve intricate spectral sequence arguments
- Careful analysis of K-theory behavior under field extensions
Cohomology Theory Developments
- New cohomology theories provide additional tools for approaching the conjecture
- (p-adic cohomology theory)
- (cohomology for varieties in positive characteristic)
- Connections to other conjectures in arithmetic geometry exploited in proof strategies
- (relating motivic cohomology to special values of L-functions)
- (relating special values of L-functions to arithmetic invariants)
Key Terms to Review (39)
Abelian varieties: Abelian varieties are a class of algebraic varieties that are also group varieties, meaning they have a group structure that is compatible with their algebraic structure. These varieties can be thought of as higher-dimensional generalizations of elliptic curves and play a central role in various areas of mathematics, particularly in number theory and algebraic geometry.
Algebraic K-Theory: Algebraic K-theory is a branch of mathematics that studies projective modules and their relationships to various algebraic structures through the lens of homotopy theory. This area of study is crucial for understanding deeper connections between algebraic objects and topological spaces, providing insights into the structure of rings and the behavior of vector bundles.
Arithmetic Geometry: Arithmetic geometry is a field that combines algebraic geometry and number theory to study the solutions of polynomial equations with a focus on their arithmetic properties. This area explores how geometric methods can yield insights into number-theoretic problems, particularly through the lens of rational and integral points on varieties. By examining these connections, arithmetic geometry bridges algebraic concepts with cohomological theories and leads to profound results regarding various conjectures and the structure of Galois cohomology.
Arithmetic schemes: Arithmetic schemes are algebraic structures that arise in the study of algebraic geometry, particularly in relation to number theory. They generalize the notion of schemes by incorporating arithmetic data, allowing for a unified approach to study both algebraic and arithmetic properties. This connection is crucial when examining the interactions between K-theory and number theory, particularly as it pertains to understanding the Quillen-Lichtenbaum conjecture.
Beilinson Conjectures: The Beilinson Conjectures are a set of conjectures in algebraic K-theory that link the ranks of the K-groups of a smooth projective variety to its Chow groups and special values of L-functions associated with its motive. They provide a framework to relate algebraic geometry and number theory, extending the classical ideas of the Birch and Swinnerton-Dyer conjecture and the conjectures of Tate and Artin.
Bloch-Kato Conjecture: The Bloch-Kato Conjecture is a significant statement in algebraic K-theory and number theory, proposing a relationship between the K-theory of a field and its Galois cohomology. This conjecture connects various mathematical areas, indicating that the K-groups of a field can be related to its étale cohomology, which is a cornerstone for understanding Galois representations.
Cyclotomic character: A cyclotomic character is a homomorphism from a Galois group to the multiplicative group of roots of unity, often denoted as $ar{Q}^{ imes}$. It captures information about the action of the Galois group on the roots of unity, particularly in relation to field extensions and number theory. This character plays a crucial role in understanding the structure of various cohomological theories, particularly in the context of K-theory and its connections to number fields.
E_2^{p,q}: The term e_2^{p,q} refers to the second page of the Adams spectral sequence, which is a tool used in stable homotopy theory and algebraic K-theory. This term plays a crucial role in connecting stable homotopy groups of spheres with various cohomological structures, revealing deep relationships between algebraic K-theory and the topology of spaces. Understanding e_2^{p,q} is essential for exploring how these structures behave under various conditions, particularly in relation to the Quillen-Lichtenbaum conjecture.
étale cohomology: Étale cohomology is a powerful tool in algebraic geometry that extends the notion of cohomology to schemes in a way that captures information about their geometric properties. It is particularly useful for studying the properties of algebraic varieties over fields, especially in the context of Galois actions and arithmetic geometry.
étale descent spectral sequence: The étale descent spectral sequence is a tool in algebraic K-theory that allows for the computation of the K-groups of schemes through their étale covers. This spectral sequence arises from the use of descent theory, specifically the idea that properties can be 'descended' from a covering space to the base space. It connects the topology of schemes with the algebraic structure of their K-groups, providing insights into how local information can inform global properties.
Finite coefficients: Finite coefficients refer to the use of a finite set of values or elements in a mathematical structure, particularly in the context of algebraic objects like groups, rings, or modules. In many cases, working with finite coefficients allows for more manageable computations and can lead to significant insights, especially when connecting algebraic structures to topological spaces or number theory.
Finite Fields: Finite fields, also known as Galois fields, are algebraic structures consisting of a finite number of elements where addition, subtraction, multiplication, and division (except by zero) are well-defined. These fields are crucial in various areas of mathematics, including coding theory and algebraic geometry, and they play a significant role in constructing and understanding other mathematical concepts, such as the Q-construction and the Quillen-Lichtenbaum conjecture.
Galois Cohomology: Galois cohomology is a branch of mathematics that studies the relationship between field extensions and the group of automorphisms of those fields, particularly through cohomological techniques. This concept connects algebra, number theory, and geometry, providing tools to understand the structure of fields and their symmetries, which are essential in various mathematical contexts.
Galois descent techniques: Galois descent techniques are methods in algebraic geometry and number theory that allow one to understand the behavior of algebraic objects under the action of Galois groups. They are particularly useful for relating properties of schemes over a base field to those over a field extension, facilitating the transfer of geometric or arithmetic information. These techniques are essential for proving results like the Quillen-Lichtenbaum conjecture, as they provide a framework to analyze how objects behave when 'descended' from a larger field to a smaller one.
Global fields: Global fields are a class of fields that encompass both number fields and function fields of one variable over a finite field. They are significant in algebraic number theory and algebraic geometry, connecting various concepts and structures, including the arithmetic of numbers and the properties of rational functions.
Higher algebraic k-theory: Higher algebraic k-theory is an extension of algebraic k-theory that deals with higher-dimensional analogues of the Grothendieck groups, specifically involving K-groups K_n associated with various algebraic structures. This theory aims to understand the relationships between algebraic varieties and their topological features through these K-groups, bridging classical algebraic geometry and modern homotopy theory. The understanding of higher k-theory plays a crucial role in analyzing motivic cohomology and in formulating conjectures like the Quillen-Lichtenbaum conjecture, which connects these theories with number theory.
Homotopy Fixed Points: Homotopy fixed points refer to a concept in algebraic topology that generalizes the idea of fixed points under group actions in a homotopical context. They arise when considering the action of a topological group on a space, leading to the notion of spaces that remain unchanged under this action up to homotopy, rather than pointwise. This concept is crucial for understanding the relationship between homotopical invariants and fixed point theories, particularly in the context of cohomology theories and their applications to stable homotopy theory.
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-spectra: K-spectra are a concept in algebraic K-theory that refers to the spectrum associated with a given ring or scheme, which serves as a tool for understanding its K-theory. This framework provides a way to study the relationships between different algebraic objects through their K-theory, ultimately connecting them to topological and geometric insights. K-spectra allow mathematicians to analyze how various structures interact within the realm of algebraic geometry 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.
L-functions: L-functions are complex functions that encode number-theoretic information and are closely linked to various areas of mathematics, including algebraic geometry and number theory. They generalize the Riemann zeta function and are crucial in understanding properties of arithmetic objects like elliptic curves and modular forms. These functions play a pivotal role in several conjectures, notably in bridging connections between different mathematical domains.
Milnor K-Theory: Milnor K-Theory is a branch of algebraic K-theory that studies the behavior of fields and schemes through the lens of higher-dimensional cohomology, particularly focusing on the K-groups associated with fields and their extensions. This theory extends classical notions of K-theory by capturing Galois cohomological information and connecting it with various areas in mathematics, including algebraic geometry and number theory.
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.
Motivic Spectral Sequence: A motivic spectral sequence is a computational tool used in algebraic geometry to derive information about the stable homotopy category of schemes, particularly in relation to motives. It connects the topology of algebraic varieties with algebraic K-theory and allows for the computation of various invariants by filtering complex structures into manageable pieces, ultimately providing insights into their relationships and properties.
Norm residue homomorphism: The norm residue homomorphism is a crucial tool in algebraic K-theory that connects Milnor K-theory with Galois cohomology. It takes elements from the Milnor K-group of a field, providing a way to translate questions about the field's residues into the context of algebraic cycles and higher K-groups, highlighting important relationships between different areas of number theory.
Number theory: Number theory is a branch of mathematics dedicated to the study of integers and their properties. It explores concepts such as divisibility, prime numbers, and congruences, serving as a foundational area that connects various mathematical disciplines including algebraic structures and analytical methods. This field plays a critical role in understanding deeper relationships in algebraic K-theory, especially through its implications in conjectures that relate algebraic and arithmetic aspects of number systems.
P-adic completion: P-adic completion is a process that transforms a mathematical object, typically a ring or a field, into a complete structure with respect to the p-adic metric. This completion allows for the rigorous handling of limits and convergence in number theory, particularly in relation to local fields. P-adic completion is essential for understanding how algebraic structures behave under the influence of prime numbers and is closely tied to many significant theorems and conjectures in algebraic geometry and number theory.
P-adic integers: P-adic integers are elements of the ring of p-adic integers, denoted as $$\mathbb{Z}_p$$, which is a system of numbers that extends the ordinary integers to include a form of 'infinitesimal' behavior with respect to a prime number p. This structure allows for a unique way of measuring distances between numbers based on divisibility by powers of p, making it useful in number theory and algebraic geometry.
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-Lichtenbaum Conjecture: The Quillen-Lichtenbaum Conjecture is a conjecture in algebraic K-theory that posits a deep connection between the K-theory of schemes over a field and the K-theory of their finite field reductions. This conjecture links various areas of mathematics, revealing how properties in algebraic K-theory can reflect geometric and topological characteristics through reductions and may also imply periodicity phenomena in K-theory.
Regulators: In the context of Algebraic K-Theory, regulators are homomorphisms that map elements from a K-group to a specific cohomology group, providing a way to measure and analyze the behavior of algebraic structures. These regulators serve as an important tool for connecting algebraic invariants with topological properties, allowing for deeper insights into the relationships between various mathematical concepts.
Rigid cohomology: Rigid cohomology is a type of cohomology theory that arises in the study of algebraic varieties over fields of positive characteristic, particularly in relation to p-adic methods. It provides a framework for understanding the behavior of geometric objects in terms of their arithmetic properties, offering connections between algebraic geometry and number theory. This concept is especially significant in contexts where traditional cohomological techniques may fail, such as in the presence of wild ramification.
Smooth Curves: Smooth curves are mathematical objects that are continuously differentiable, meaning they can be represented by polynomial or differentiable functions without any sharp corners or cusps. In the context of algebraic geometry and K-theory, smooth curves play a vital role as they often serve as the building blocks for more complex varieties, influencing various properties such as their cohomology and behavior under morphisms.
Smooth varieties: Smooth varieties are geometric objects in algebraic geometry that possess desirable properties, such as having no singular points, which means they can be described by polynomials without any abrupt changes in behavior. These varieties are essential for understanding various deep results in algebraic K-theory, especially as they facilitate connections between algebraic and topological properties. The smoothness of a variety is crucial when discussing the Merkurjev-Suslin theorem and the Quillen-Lichtenbaum conjecture, as it ensures the existence of certain structures that allow for meaningful K-theoretic calculations.
Spectral sequence: A spectral sequence is a mathematical tool used to compute homology or cohomology groups by organizing data into a sequence of pages that converge to the desired result. This method allows for the systematic handling of complex calculations by breaking them down into simpler, more manageable pieces, each represented on different pages. Spectral sequences are particularly powerful in algebraic topology and algebraic K-theory, where they help analyze various structures and relationships within these fields.
Stable homotopy category: The stable homotopy category is a framework in algebraic topology that captures the idea of stable phenomena by identifying spaces and spectra up to stable equivalence. It serves as a fundamental setting for studying stable homotopy theory, where one focuses on properties that remain invariant under suspension, allowing for a more manageable analysis of complex topological structures. This category is crucial in linking various mathematical concepts such as K-theory and cohomology theories.
Syntomic Cohomology: Syntomic cohomology is a type of cohomology theory in algebraic geometry that generalizes the concept of étale cohomology, particularly in relation to schemes over a base field. It plays a crucial role in understanding the behavior of algebraic objects under various morphisms, and it connects deeply with notions of p-adic Hodge theory and the study of the Galois representations associated with schemes.
Tamagawa number conjecture: The Tamagawa number conjecture is a hypothesis in number theory and algebraic geometry that connects the arithmetic of abelian varieties with the behavior of their Galois representations. It asserts that there is a profound relationship between the Tamagawa numbers, which measure the local contributions to the number of points on an abelian variety over finite fields, and the Galois cohomology groups associated with these varieties.
Tate Twists: Tate twists are a construction in the field of algebraic K-theory, specifically involving a certain type of twisting operation applied to K-groups. These twists are essential for understanding the behavior of K-theory under various base changes and play a crucial role in the formulation of conjectures related to the arithmetic of schemes. They allow mathematicians to connect different areas of algebraic geometry and number theory by introducing additional structure that can alter the properties of K-theory spectra.