🪡K-Theory Unit 13 – Higher Algebraic K–Theory and its Applications

Foundations of K-Theory

• K-theory studies vector bundles and projective modules over rings, providing a powerful algebraic invariant
• Originated from the study of vector bundles in topology and has since expanded to encompass a wide range of mathematical disciplines
• Topological K-theory, introduced by Alexander Grothendieck, classifies vector bundles over topological spaces
  • Assigns an abelian group to each topological space, capturing essential information about vector bundles
• Algebraic K-theory, developed by Daniel Quillen, extends the ideas of topological K-theory to the realm of rings and algebras
  • Associates a sequence of abelian groups (K-groups) to each ring, encoding algebraic and geometric properties
• K-theory has deep connections to various branches of mathematics, including algebraic geometry, number theory, and operator algebras
• Serves as a unifying language, bridging the gap between seemingly disparate mathematical concepts and structures

Introduction to Higher Algebraic K-Theory

• Higher algebraic K-theory extends the classical algebraic K-theory beyond the first two K-groups ($K_0$ and $K_1$)
• Provides a more refined and detailed understanding of the algebraic and geometric properties of rings and schemes
• Constructed using sophisticated techniques from homotopy theory and category theory
  • Involves the study of higher categories, such as $\infty$-categories and stable $\infty$-categories
• Higher K-groups ($K_n$ for $n \geq 2$) capture intricate information about the structure of rings and their modules
  • For example, $K_2$ is related to the Brauer group and the study of central simple algebras
• Plays a crucial role in the formulation and proof of important conjectures in algebraic geometry and number theory
  • Notably, the Quillen-Lichtenbaum conjecture relates higher algebraic K-theory to étale cohomology

Key Concepts and Definitions

• Exact categories: Additive categories with a distinguished class of short exact sequences, serving as a foundation for K-theory constructions
• Quillen's Q-construction: A functorial way to associate a topological space (or simplicial set) to an exact category, enabling the definition of higher K-groups
• Waldhausen categories: Categories with cofibrations and weak equivalences, generalizing exact categories and allowing for more flexible K-theory constructions
• Spectra: Infinite sequences of pointed spaces with structure maps, used to represent generalized cohomology theories like K-theory
  • K-theory spectrum: The spectrum associated with the K-theory of a ring or scheme, obtained by applying the Q-construction iteratively
• Localization theorems: Results that relate the K-theory of a scheme to its localization at various subsets (e.g., the complement of a closed subscheme)
• Milnor K-theory: A simpler version of K-theory, defined using generators and relations, that coincides with algebraic K-theory in certain cases (e.g., for fields)

Constructing Higher K-Groups

• The Q-construction, introduced by Daniel Quillen, is a fundamental tool for constructing higher K-groups
  • Assigns a topological space (or simplicial set) $\mathbf{Q}(\mathcal{C})$ to an exact category $\mathcal{C}$
  • The n-th K-group $K_n(\mathcal{C})$ is defined as the n-th homotopy group of $\mathbf{Q}(\mathcal{C})$
• The S-construction, developed by Waldhausen, extends the Q-construction to categories with cofibrations and weak equivalences
  • Allows for the construction of K-theory in more general settings, such as the K-theory of spaces or the K-theory of ring spectra
• The +-construction is another approach to defining higher K-groups, using the +-construction on the classifying space of the general linear group
  • Equivalent to the Q-construction for regular rings, but may differ in general
• Iterating the Q-construction (or S-construction) yields a spectrum, the K-theory spectrum, whose homotopy groups are the higher K-groups
• The K-theory of a scheme can be defined using the Q-construction applied to the exact category of locally free sheaves or the category of perfect complexes

Computational Techniques

• Spectral sequences are powerful tools for computing higher K-groups
  • The Atiyah-Hirzebruch spectral sequence relates the K-theory of a scheme to its cohomology theories (e.g., motivic cohomology, étale cohomology)
  • The Quillen spectral sequence relates the K-theory of a ring to its cyclic homology
• Descent techniques, such as the Nisnevich descent or étale descent, allow for the computation of K-theory by reducing to simpler cases (e.g., local rings or separably closed fields)
• The Bloch-Lichtenbaum spectral sequence provides a connection between the K-theory of a field and its motivic cohomology, enabling computations in certain cases
• Trace methods, involving the study of trace maps from K-theory to other invariants (e.g., topological cyclic homology), can yield information about K-groups
• Comparison theorems, such as the Quillen-Suslin theorem or the Gersten conjecture, relate the K-theory of a scheme to its local or geometric properties

Properties and Theorems

• Functoriality: K-theory is a contravariant functor from the category of rings (or schemes) to the category of spectra
  • Maps between rings (or schemes) induce maps between their K-theory spectra in the opposite direction
• Homotopy invariance: The K-theory of a ring is invariant under polynomial extensions
  • For a ring $R$, the natural map $K(R) \to K(R[t])$ is a homotopy equivalence
• Localization theorems: For a scheme $X$ and a closed subscheme $Y$, there is a long exact sequence relating the K-theory of $X$, $Y$, and the open complement $U = X \setminus Y$
• Projective bundle formula: For a scheme $X$ and a vector bundle $\mathcal{E}$ on $X$, the K-theory of the projective bundle $\mathbb{P}(\mathcal{E})$ is determined by the K-theory of $X$ and the powers of the tautological line bundle
• Fundamental theorems: The fundamental theorems of K-theory, such as the Fundamental Theorem of Algebraic K-Theory, provide a deeper understanding of the structure of K-groups and their relation to other invariants

Applications in Mathematics

• Algebraic geometry: K-theory is a powerful tool for studying the structure of algebraic varieties and schemes
  • The K-theory of a scheme encodes information about its vector bundles, coherent sheaves, and perfect complexes
  • K-theory plays a role in the formulation and proof of important conjectures, such as the Bloch-Kato conjecture and the Beilinson-Soulé vanishing conjecture
• Number theory: K-theory has significant applications in number theory, particularly in the study of special values of L-functions and the formulation of conjectures relating them to cohomological invariants
  • The Beilinson conjectures relate the K-theory of a variety over a number field to its L-functions and motivic cohomology
  • The Lichtenbaum conjecture connects the K-theory of a number field to its étale cohomology and Dedekind zeta function
• Topology: K-theory serves as a bridge between algebraic and topological invariants
  • The Baum-Connes conjecture relates the K-theory of group C*-algebras to the topology of classifying spaces
  • The Farrell-Jones conjecture describes the K-theory of group rings in terms of the topology of certain classifying spaces
• Noncommutative geometry: K-theory is a central tool in noncommutative geometry, where it is used to study noncommutative spaces and their properties
  • The K-theory of C*-algebras provides a powerful invariant for classifying noncommutative spaces
  • Cyclic homology theories, closely related to K-theory, are used to formulate and study noncommutative analogues of classical geometric concepts

Connections to Other Fields

• Homotopy theory: Higher algebraic K-theory is deeply intertwined with homotopy theory, as its construction relies on techniques from stable homotopy theory and higher category theory
  • K-theory spectra are objects of study in stable homotopy theory, and their properties are often investigated using tools from this field
• Operator algebras: The K-theory of C*-algebras and von Neumann algebras plays a significant role in the classification and study of these operator algebras
  • The K₀-group of a C*-algebra is a crucial invariant in the classification of AF (approximately finite-dimensional) algebras
  • The Baum-Connes conjecture, formulated in terms of K-theory, has important implications for the structure and properties of group C*-algebras
• Algebraic topology: K-theory has its roots in algebraic topology, and there are deep connections between the two fields
  • The Atiyah-Hirzebruch spectral sequence relates the K-theory of a space to its ordinary cohomology
  • The Adams operations, which are ring homomorphisms on K-theory groups, have analogues in other cohomology theories and provide a link between K-theory and stable homotopy theory
• Mathematical physics: K-theory has found applications in various areas of mathematical physics, particularly in the study of quantum field theories and string theory
  • The K-theory of C*-algebras is used in the classification of D-brane charges in string theory
  • K-theoretic invariants, such as the Fredholm index, appear in the study of elliptic operators and their role in quantum field theories
• Representation theory: K-theory is closely related to representation theory, as the K-theory of group rings and related algebras encodes information about their representation-theoretic properties
  • The Baum-Connes conjecture, which involves the K-theory of group C*-algebras, has implications for the representation theory of discrete groups
  • The K-theory of Hecke algebras and affine Hecke algebras plays a role in the study of representations of p-adic groups and quantum groups