🪡K-Theory Unit 14 – K–Theory in Algebraic Geometry and Number Theory

Foundations of K-Theory

• K-theory studies invariants of mathematical objects (rings, schemes, varieties) using techniques from algebraic topology and category theory
• Originated from Alexander Grothendieck's work on coherent sheaves and the Grothendieck group in the 1950s
• Developed further by Hyman Bass, Michael Atiyah, and Friedrich Hirzebruch in the 1960s
• Connects various branches of mathematics (algebraic geometry, number theory, topology, representation theory)
• Provides a unified framework for studying vector bundles, projective modules, and algebraic cycles
  • Vector bundles generalize the concept of a vector space over a manifold or algebraic variety
  • Projective modules are a generalization of free modules and play a crucial role in non-commutative ring theory
• K-groups are abelian groups associated with a mathematical object that capture its essential properties and structure
• Higher K-groups ($K_n$ for $n > 0$) contain more refined information about the object and its symmetries

Key Concepts and Definitions

• Grothendieck group $K_0(X)$ of a scheme $X$ is the free abelian group generated by coherent sheaves on $X$ modulo the relation $[F] = [F'] + [F'']$ for every short exact sequence $0 \to F' \to F \to F'' \to 0$
• Higher algebraic K-groups $K_n(R)$ of a ring $R$ are defined using the Quillen's Q-construction or the Waldhausen's S-construction
  • $K_1(R)$ is closely related to the group of units $R^\times$ and the Picard group $\mathrm{Pic}(R)$
  • $K_2(R)$ is related to the Brauer group $\mathrm{Br}(R)$ and the Milnor K-theory
• Bott periodicity establishes a periodic relationship between the K-groups of a topological space $X$: $K_n(X) \cong K_{n+2}(X)$ for all $n \geq 0$
• Chern character is a ring homomorphism from the K-theory of a smooth variety $X$ to its Chow groups tensored with $\mathbb{Q}$: $\mathrm{ch}: K_0(X) \to \bigoplus_{i \geq 0} \mathrm{CH}^i(X) \otimes \mathbb{Q}$
• Adams operations $\psi^k$ are ring endomorphisms of K-theory that generalize the k-th power map on line bundles and provide a way to study the structure of K-groups
• Quillen's higher algebraic K-theory extends the notion of K-theory to exact categories and provides a powerful tool for studying the K-groups of schemes and rings

Algebraic K-Groups

• $K_0(X)$ of a scheme $X$ classifies coherent sheaves on $X$ up to stable equivalence
  • For a smooth variety $X$, $K_0(X)$ is isomorphic to the Grothendieck group of vector bundles on $X$
• $K_1(R)$ of a ring $R$ is the abelianization of the infinite general linear group $\mathrm{GL}(R)$
  • For a field $F$, $K_1(F)$ is isomorphic to the multiplicative group $F^\times$
  • $K_1(R)$ captures information about the automorphisms of projective modules over $R$
• $K_2(R)$ is related to the Steinberg group $\mathrm{St}(R)$ and the Milnor K-theory $K_2^M(R)$
  • For a field $F$, there is a surjective homomorphism $K_2(F) \to K_2^M(F)$ whose kernel is the group of universal symbols
• Higher K-groups $K_n(R)$ for $n > 2$ are defined using the Quillen's Q-construction or the Waldhausen's S-construction and have a rich structure
  • They are related to the motivic cohomology groups and the Bloch's higher Chow groups
• Milnor K-theory $K_*^M(F)$ of a field $F$ is a graded ring defined using the tensor algebra of $F^\times$ modulo the Steinberg relations
  • There is a natural homomorphism from Milnor K-theory to Quillen's algebraic K-theory: $K_n^M(F) \to K_n(F)$ for all $n \geq 0$

K-Theory in Algebraic Geometry

• K-theory provides a powerful tool for studying algebraic cycles and motives in algebraic geometry
• For a smooth projective variety $X$, the Grothendieck group $K_0(X)$ is isomorphic to the Chow ring $\mathrm{CH}^*(X) \otimes \mathbb{Z}$
  • The Chern character gives a ring isomorphism $K_0(X) \otimes \mathbb{Q} \cong \mathrm{CH}^*(X) \otimes \mathbb{Q}$
• Algebraic K-theory of a scheme $X$ is related to its motivic cohomology groups $H^{p,q}(X, \mathbb{Z})$ via the motivic spectral sequence
  • The motivic cohomology groups generalize the Chow groups and provide a way to study the arithmetic and geometric properties of $X$
• Bloch's higher Chow groups $\mathrm{CH}^p(X, n)$ are a generalization of the usual Chow groups that capture information about the algebraic cycles on $X$
  • There is a natural homomorphism from the higher Chow groups to the algebraic K-groups: $\mathrm{CH}^p(X, n) \to K_n(X)^{(p)}$ for all $p, n \geq 0$
• Riemann-Roch theorem for algebraic K-theory relates the Euler characteristic of a coherent sheaf on a smooth projective variety $X$ to its Chern character in the Chow ring of $X$
• Quillen-Lichtenbaum conjecture relates the algebraic K-theory of a scheme $X$ to its étale cohomology groups with finite coefficients
  • It provides a connection between the algebraic and topological properties of $X$ and has important applications in arithmetic geometry

K-Theory in Number Theory

• Algebraic K-theory of rings of integers and their quotients plays a crucial role in number theory
• For a number field $F$, the K-groups $K_n(\mathcal{O}_F)$ of its ring of integers $\mathcal{O}_F$ contain important arithmetic information
  • $K_0(\mathcal{O}_F)$ is the Picard group of $\mathcal{O}_F$ and classifies the fractional ideals of $F$ up to principal ideals
  • $K_1(\mathcal{O}_F)$ is the group of units $\mathcal{O}_F^\times$ and is related to the class number of $F$
  • Higher K-groups $K_n(\mathcal{O}_F)$ for $n > 1$ are related to the Dedekind zeta function and the Tamagawa number of $F$
• Quillen-Lichtenbaum conjecture for rings of integers relates the K-groups $K_n(\mathcal{O}_F)$ to the étale cohomology groups of $\mathrm{Spec}(\mathcal{O}_F)$ with finite coefficients
  • It provides a way to study the arithmetic properties of $F$ using the tools of algebraic geometry and topology
• Beilinson conjectures relate the values of the Dedekind zeta function $\zeta_F(s)$ at integers $s \leq 0$ to the K-groups and the motivic cohomology groups of $F$
  • They provide a deep connection between the analytic and algebraic properties of number fields and have important applications in the study of special values of L-functions
• Iwasawa theory studies the behavior of arithmetic objects (class groups, units, Galois modules) in towers of number fields using the tools of algebraic K-theory and p-adic analysis
  • Main conjecture of Iwasawa theory relates the p-adic L-functions to the characteristic ideals of certain Iwasawa modules and has important applications in the study of cyclotomic fields and elliptic curves

Applications and Examples

• Algebraic K-theory has applications in various branches of mathematics, including algebraic geometry, number theory, topology, and representation theory
• In algebraic geometry, K-theory is used to study algebraic cycles, motives, and the arithmetic properties of schemes
  • Example: Grothendieck's proof of the Grothendieck-Riemann-Roch theorem uses the K-theory of coherent sheaves and the Chern character
• In number theory, K-theory is used to study the arithmetic properties of rings of integers, special values of L-functions, and the Galois modules
  • Example: Quillen's computation of the K-groups of finite fields using the Adams operations and the Brauer lift
• In topology, K-theory is used to study the classification of vector bundles, the Atiyah-Singer index theorem, and the Novikov conjecture
  • Example: Atiyah-Hirzebruch spectral sequence relates the K-theory of a topological space to its singular cohomology
• In representation theory, K-theory is used to study the representation rings of groups and the character formulas
  • Example: Weyl character formula expresses the characters of irreducible representations of compact Lie groups in terms of the K-theory of flag varieties
• Other examples include:
  • Milnor's construction of the K-theory of fields and its relation to the quadratic forms and the Witt rings
  • Suslin's proof of the Quillen-Lichtenbaum conjecture for fields using the motivic cohomology and the Bloch-Kato conjecture
  • Voevodsky's proof of the Milnor conjecture on the Galois cohomology of fields using the motivic cohomology and the norm residue isomorphism theorem

Advanced Topics and Current Research

• Motivic homotopy theory is a generalization of algebraic K-theory that studies the homotopy theory of schemes and motives using the language of model categories and infinity-categories
  • Voevodsky's construction of the motivic stable homotopy category $\mathbf{SH}(k)$ over a field $k$ provides a powerful tool for studying the motivic cohomology and the algebraic K-theory of $k$
  • Motivic Adams spectral sequence relates the motivic stable homotopy groups to the motivic cohomology groups and the algebraic K-groups
• Trace methods in algebraic K-theory use the trace maps from the K-theory to the cyclic homology and the topological cyclic homology to study the K-groups of rings and schemes
  • Topological cyclic homology $\mathrm{TC}(R)$ of a ring $R$ is a refinement of the Hochschild and cyclic homology that captures the arithmetic properties of $R$
  • Trace methods have important applications in the study of the K-theory of local fields and the crystalline cohomology of schemes
• Equivariant K-theory studies the K-theory of schemes and stacks with group actions using the tools of equivariant homotopy theory and representation theory
  • Atiyah-Segal completion theorem relates the equivariant K-theory of a compact Lie group $G$ to the representation ring of $G$
  • Equivariant K-theory has important applications in the study of the K-theory of quotient stacks and the geometric representation theory
• K-theory of non-commutative rings and categories is a generalization of the classical K-theory that studies the K-groups of non-commutative rings, exact categories, and triangulated categories
  • Waldhausen's S-construction provides a way to define the K-theory of categories with cofibrations and weak equivalences
  • K-theory of non-commutative rings has important applications in the study of the K-theory of group rings, the cyclic homology of algebras, and the classification of C*-algebras
• Current research in algebraic K-theory focuses on the following topics:
  • Computation of the K-groups of rings and schemes using the trace methods, the motivic homotopy theory, and the equivariant techniques
  • Study of the arithmetic properties of the K-groups of rings of integers and their relation to the special values of L-functions and the Galois modules
  • Applications of K-theory to the study of the motivic cohomology, the algebraic cycles, and the motives in algebraic geometry
  • Generalization of K-theory to the non-commutative settings, such as the K-theory of dg-categories, the K-theory of ring spectra, and the K-theory of stable infinity-categories

Problem-Solving Techniques

• Computation of K-groups using the Quillen's Q-construction or the Waldhausen's S-construction
  • Example: Compute the K-groups of a finite field $\mathbb{F}_q$ using the Quillen's Q-construction and the Adams operations
• Use of the long exact sequences and the spectral sequences to compute the K-groups of schemes and rings
  • Example: Use the localization sequence to compute the K-groups of a scheme $X$ in terms of the K-groups of its closed subschemes and their complements
• Application of the Chern character and the Adams operations to study the structure of K-groups
  • Example: Use the Chern character to decompose the K-theory of a smooth projective variety into the direct sum of its Chow groups tensored with $\mathbb{Q}$
• Use of the trace methods and the descent techniques to compute the K-groups of rings and schemes
  • Example: Use the trace map from the K-theory to the topological cyclic homology to compute the K-groups of a local field
• Reduction of the computation of K-groups to the study of the motivic cohomology and the algebraic cycles using the motivic spectral sequence
  • Example: Use the motivic spectral sequence to express the K-groups of a smooth projective variety in terms of its motivic cohomology groups and the Adams eigenspaces
• Use of the equivariant techniques and the representation theory to study the K-theory of group actions and quotient stacks
  • Example: Use the Atiyah-Segal completion theorem to compute the equivariant K-theory of a compact Lie group in terms of its representation ring
• Application of the categorical and the homotopical methods to study the K-theory of non-commutative rings and categories
  • Example: Use the Waldhausen's S-construction to define the K-theory of a triangulated category and study its properties using the tools of homotopy theory
• Use of the computational tools, such as the computer algebra systems and the programming languages, to perform explicit calculations and verify conjectures
  • Example: Use the SageMath or the Macaulay2 to compute the K-groups of a given ring or scheme and compare the results with the theoretical predictions