🔢Algebraic K-Theory Unit 10 – Serre and Quillen-Lichtenbaum Conjectures

Historical Context

• Algebraic K-theory emerged in the 1950s and 1960s as a generalization of classical K-theory, which studies vector bundles on topological spaces
• Early work by Alexander Grothendieck and Michael Atiyah laid the foundations for the development of algebraic K-theory
  • Grothendieck introduced the concept of K-groups for categories, extending the notion of K-theory to algebraic geometry
  • Atiyah's work on the K-theory of coherent sheaves provided a framework for studying algebraic vector bundles
• Hyman Bass and Daniel Quillen made significant contributions to the field in the 1970s
  • Bass developed the theory of algebraic K-theory for rings, introducing higher K-groups and establishing fundamental properties
  • Quillen's work on the higher algebraic K-theory of rings and schemes revolutionized the field and led to the formulation of the Quillen-Lichtenbaum conjecture
• The Serre conjecture, proposed by Jean-Pierre Serre in the 1960s, played a crucial role in motivating the development of algebraic K-theory
  • Serre's conjecture related the Galois cohomology of a field to its algebraic K-theory, suggesting deep connections between arithmetic and geometry
• The resolution of the Quillen-Lichtenbaum conjecture in the 1990s and early 2000s marked a major milestone in algebraic K-theory
  • The conjecture provided a link between algebraic K-theory and étale cohomology, bridging the gap between algebraic and topological invariants

Foundations of Algebraic K-Theory

• Algebraic K-theory associates K-groups $K_i(R)$ to a ring $R$, generalizing the construction of the Grothendieck group $K_0(R)$
  • $K_0(R)$ is the abelian group generated by isomorphism classes of finitely generated projective $R$-modules, with relations given by short exact sequences
  • Higher K-groups $K_i(R)$ for $i > 0$ are defined using the Quillen Q-construction or the Waldhausen S-construction
• The K-groups of a ring $R$ encode important arithmetic and geometric information about $R$
  • $K_0(R)$ captures information about the projective modules over $R$ and the Picard group of $R$
  • $K_1(R)$ is related to the group of units of $R$ and the Whitehead group of $R$
  • Higher K-groups $K_i(R)$ for $i > 1$ are more mysterious and are the subject of active research
• Algebraic K-theory satisfies various functorial properties, such as covariance with respect to ring homomorphisms and contravariance with respect to certain functors
  • The covariance property allows for the study of K-theory in families and the construction of K-theory spectra
  • The contravariance property is exemplified by the long exact localization sequence, relating the K-theory of a ring to that of a quotient ring
• The K-theory of a scheme $X$ is defined by considering the K-theory of the category of vector bundles or coherent sheaves on $X$
  • The K-theory of a scheme encodes information about the geometry and arithmetic of $X$
  • The K-theory of a smooth projective variety over a field is closely related to its Chow groups and cycle class maps

Serre's Conjecture: Key Ideas

• Serre's conjecture relates the Galois cohomology of a field $k$ to its algebraic K-theory
  • The conjecture states that for a field $k$ and a prime $\ell$ different from the characteristic of $k$, the Galois cohomology groups $H^i(k, \mathbb{Z}_\ell(n))$ are isomorphic to the étale cohomology groups $H^i_{ét}(\text{Spec}(k), \mathbb{Z}_\ell(n))$
  • The conjecture suggests a deep connection between arithmetic (Galois cohomology) and geometry (algebraic K-theory)
• The conjecture is motivated by the analogy between the K-theory of a field and its Galois cohomology
  • The K-groups of a field $k$ can be interpreted as the homotopy groups of a certain spectrum $K(k)$, called the K-theory spectrum of $k$
  • The Galois cohomology groups of $k$ with coefficients in $\mathbb{Z}_\ell(n)$ can be interpreted as the homotopy groups of the étale homotopy type of $\text{Spec}(k)$
• The Bloch-Kato conjecture, a generalization of Serre's conjecture, relates the Galois cohomology of a field to its Milnor K-theory
  • Milnor K-theory is a simpler version of algebraic K-theory that is easier to compute and study
  • The Bloch-Kato conjecture provides a more accessible approach to understanding the relationship between Galois cohomology and algebraic K-theory
• The Serre conjecture has been proved in various special cases, such as for fields of cohomological dimension at most 2 and for certain classes of fields (p-adic fields, global fields)
  • The proofs often rely on the computation of the K-theory and Galois cohomology of the fields in question and the construction of explicit isomorphisms between them

Quillen-Lichtenbaum Conjecture: Overview

• The Quillen-Lichtenbaum conjecture is a generalization of Serre's conjecture to the setting of algebraic K-theory and étale cohomology
  • The conjecture states that for a scheme $X$ and a prime $\ell$ invertible on $X$, there is a natural isomorphism between the étale cohomology groups $H^i_{ét}(X, \mathbb{Z}_\ell(n))$ and the $\ell$-adic algebraic K-theory groups $K_{i}(X)_{\mathbb{Z}_\ell}$ for sufficiently large $i$ and $n$
  • The isomorphism is expected to be compatible with the product structures on both sides and with the action of the absolute Galois group of the base field
• The conjecture provides a link between the algebraic K-theory of a scheme and its étale cohomology, bridging the gap between algebraic and topological invariants
  • Étale cohomology is a cohomology theory for schemes that takes into account the arithmetic and geometric properties of the scheme
  • The conjecture suggests that the algebraic K-theory of a scheme, which encodes information about vector bundles and coherent sheaves, is closely related to its étale cohomology
• The Quillen-Lichtenbaum conjecture has important consequences for the study of special values of L-functions and the Beilinson conjectures
  • The Beilinson conjectures relate the special values of L-functions to the K-theory and motivic cohomology of algebraic varieties
  • The Quillen-Lichtenbaum conjecture provides a way to compare the K-theory and étale cohomology of varieties, which is crucial for understanding the Beilinson conjectures
• The conjecture has been proved in various special cases, such as for schemes over finite fields and for certain classes of varieties (curves, surfaces)
  • The proofs often rely on the computation of the K-theory and étale cohomology of the schemes in question and the construction of explicit isomorphisms between them
  • The Voevodsky-Rost theorem, proving the Bloch-Kato conjecture for Milnor K-theory, has been a key ingredient in many proofs of the Quillen-Lichtenbaum conjecture

Connections to Étale Cohomology

• Étale cohomology is a cohomology theory for schemes that takes into account the arithmetic and geometric properties of the scheme
  • It is defined using the étale topology, a Grothendieck topology on the category of schemes that captures information about finite covers and separable field extensions
  • Étale cohomology groups $H^i_{ét}(X, \mathcal{F})$ are defined for a scheme $X$ and an étale sheaf $\mathcal{F}$ on $X$, generalizing the notion of singular cohomology for topological spaces
• The Quillen-Lichtenbaum conjecture relates the algebraic K-theory of a scheme to its étale cohomology with coefficients in the twist $\mathbb{Z}_\ell(n)$
  • The twist $\mathbb{Z}_\ell(n)$ is an étale sheaf that plays a role analogous to the constant sheaf $\mathbb{Z}$ in singular cohomology
  • The conjecture suggests that the algebraic K-theory groups $K_i(X)$ contain information about the étale cohomology groups $H^i_{ét}(X, \mathbb{Z}_\ell(n))$ for sufficiently large $i$ and $n$
• Étale cohomology satisfies various functorial properties, such as covariance with respect to morphisms of schemes and the existence of long exact sequences
  • These properties are crucial for the study of étale cohomology and its relationship to algebraic K-theory
  • The functorial properties of étale cohomology are compatible with those of algebraic K-theory, allowing for the comparison of the two theories
• The Hochschild-Serre spectral sequence relates the étale cohomology of a scheme to the étale cohomology of its base change to a separable closure of the base field
  • This spectral sequence is a key tool in the study of étale cohomology and its relationship to Galois cohomology
  • The Hochschild-Serre spectral sequence is compatible with the Atiyah-Hirzebruch spectral sequence in algebraic K-theory, providing a way to compare the two theories
• The proof of the Quillen-Lichtenbaum conjecture often involves the construction of explicit isomorphisms between the algebraic K-theory and étale cohomology of a scheme
  • These isomorphisms are typically constructed using the Atiyah-Hirzebruch and Hochschild-Serre spectral sequences, along with other tools from algebraic geometry and homotopy theory
  • The compatibility of these spectral sequences with the product structures on K-theory and étale cohomology is crucial for establishing the desired isomorphisms

Proof Strategies and Major Developments

• The proof of the Quillen-Lichtenbaum conjecture has been a major goal in algebraic K-theory and has led to the development of various proof strategies and techniques
  • The conjecture has been proved in various special cases, such as for schemes over finite fields and for certain classes of varieties (curves, surfaces)
  • The proofs often rely on a combination of techniques from algebraic geometry, homotopy theory, and arithmetic geometry
• One key strategy in proving the Quillen-Lichtenbaum conjecture is to use the Atiyah-Hirzebruch spectral sequence in algebraic K-theory
  • The Atiyah-Hirzebruch spectral sequence relates the algebraic K-theory of a scheme to its motivic cohomology, which is a cohomology theory that combines algebraic K-theory and Chow groups
  • The spectral sequence provides a way to compare the algebraic K-theory and étale cohomology of a scheme, using the relationship between motivic cohomology and étale cohomology
• Another important technique is the use of the Beilinson-Lichtenbaum conjecture, which relates the motivic cohomology of a scheme to its étale cohomology
  • The Beilinson-Lichtenbaum conjecture is a generalization of the Quillen-Lichtenbaum conjecture and provides a more direct link between motivic cohomology and étale cohomology
  • The conjecture has been proved in various special cases and has been a key ingredient in many proofs of the Quillen-Lichtenbaum conjecture
• The Voevodsky-Rost theorem, proving the Bloch-Kato conjecture for Milnor K-theory, has been another crucial development in the study of the Quillen-Lichtenbaum conjecture
  • The Bloch-Kato conjecture relates the Galois cohomology of a field to its Milnor K-theory, which is a simpler version of algebraic K-theory
  • The Voevodsky-Rost theorem provides a way to compare Milnor K-theory and étale cohomology, which can be used to prove the Quillen-Lichtenbaum conjecture in certain cases
• The proof of the Quillen-Lichtenbaum conjecture for schemes over finite fields by Suslin and Voevodsky in the 1990s was a major breakthrough
  • Their proof used a combination of techniques from algebraic geometry, homotopy theory, and arithmetic geometry, including the Beilinson-Lichtenbaum conjecture and the Voevodsky-Rost theorem
  • The proof showcased the power of these new techniques and paved the way for further progress on the conjecture

Applications in Number Theory

• The Quillen-Lichtenbaum conjecture has important applications in number theory, particularly in the study of special values of L-functions and the Beilinson conjectures
  • The Beilinson conjectures relate the special values of L-functions to the K-theory and motivic cohomology of algebraic varieties
  • The Quillen-Lichtenbaum conjecture provides a way to compare the K-theory and étale cohomology of varieties, which is crucial for understanding the Beilinson conjectures
• The conjecture has implications for the study of the Birch and Swinnerton-Dyer conjecture, which relates the rank of the Mordell-Weil group of an elliptic curve to the order of vanishing of its L-function at $s=1$
  • The Birch and Swinnerton-Dyer conjecture can be reformulated in terms of the K-theory and étale cohomology of the elliptic curve
  • The Quillen-Lichtenbaum conjecture provides a way to compare these invariants and study the Birch and Swinnerton-Dyer conjecture from a new perspective
• The conjecture also has applications to the study of the Tate conjecture, which relates the Picard group of a variety to its étale cohomology
  • The Tate conjecture can be reformulated in terms of the K-theory and étale cohomology of the variety
  • The Quillen-Lichtenbaum conjecture provides a way to compare these invariants and study the Tate conjecture from a new perspective
• The conjecture has been used to prove results about the K-theory of number fields and their rings of integers
  • For example, the conjecture has been used to compute the K-theory of the ring of integers of a number field in terms of its étale cohomology
  • These computations have important implications for the study of class groups and unit groups of number fields
• The Quillen-Lichtenbaum conjecture has also been used to study the relationship between the K-theory of a variety and its Chow groups
  • The Chow groups of a variety are important invariants that encode information about the cycles on the variety
  • The conjecture provides a way to compare the K-theory and Chow groups of a variety, leading to new insights into the geometry of the variety

Open Questions and Future Directions

• Despite significant progress, the Quillen-Lichtenbaum conjecture remains open in full generality
  • The conjecture has been proved in various special cases, but a complete proof for all schemes and all coefficients is still unknown
  • Proving the conjecture in full generality is a major open problem in algebraic K-theory and is expected to require new ideas and techniques
• One direction for future research is to explore the relationship between the Quillen-Lichtenbaum conjecture and other conjectures in algebraic K-theory and number theory
  • For example, the Beilinson-Soulé conjecture,