🪡K-Theory Unit 12 – Introduction to Algebraic K–Theory

Key Concepts and Definitions

• Algebraic K-theory studies invariants of rings, schemes, and categories using tools from algebraic topology and homotopy theory
• Grothendieck group $K_0(C)$ constructed from a category $C$ with exact sequences, generalizing the notion of the ideal class group of a ring
• Higher K-groups $K_n(R)$ defined for a ring $R$ using the classifying space of the general linear group $BGL(R)$ or the plus construction $BGL(R)^+$
• Quillen's Q-construction and plus construction provide alternative definitions of higher K-groups with better functorial properties
• Milnor K-theory defined using tensor algebras and symbolic powers of the multiplicative group, related to motivic cohomology
  • Captures information about the structure of the multiplicative group of a field
• Negative K-groups $K_{-n}(R)$ extend the theory to negative indices, related to the chromatic filtration in stable homotopy theory
• K-theory spectra constructed using infinite loop space machines (Segal's $\Gamma$-spaces or Waldhausen's S-construction) to study the homotopy groups of K-theory

Historical Context and Development

• Algebraic K-theory originated in the late 1950s with the work of Alexander Grothendieck on the Grothendieck group and Grothendieck-Riemann-Roch theorem
• Hyman Bass and Daniel Quillen developed the foundations of higher algebraic K-theory in the late 1960s and early 1970s
  • Bass defined K1 and K2 using general linear groups and Steinberg symbols
  • Quillen introduced the plus construction and the Q-construction for defining higher K-groups
• Quillen's work on the relationship between K-theory and cohomology theories (Quillen-Lichtenbaum conjecture) spurred further developments
• Friedhelm Waldhausen introduced a general framework for K-theory of categories with cofibrations and weak equivalences in the 1980s
• Vladimir Voevodsky's work on motivic cohomology and the Milnor conjecture in the 1990s led to new connections between K-theory and algebraic geometry
• Recent developments include trace methods, topological cyclic homology, and the study of K-theory of derived categories and stable $\infty$-categories

Fundamental Groups and Categories

• K-theory is built on the foundations of category theory, with key examples being the category of finitely generated projective modules over a ring and the category of vector bundles on a topological space
• Exact categories (categories with a notion of short exact sequences) provide a general framework for constructing K-theory
• Waldhausen categories (categories with cofibrations and weak equivalences) allow for the construction of K-theory spectra
• The fundamental groupoid of a topological space (category with objects as points and morphisms as homotopy classes of paths) encodes information about its K-theory
• The Quillen K-theory of a ring $R$ is related to the algebraic K-theory of the category of finitely generated projective $R$-modules
• The K-theory of a scheme $X$ is defined using the category of locally free sheaves on $X$ or the category of perfect complexes on $X$
• Higher categories (2-categories, $\infty$-categories) provide a framework for studying K-theory in more general settings

K0 Groups and Their Properties

• The zeroth K-group $K_0(C)$ of a category $C$ with exact sequences is the Grothendieck group of $C$, defined as the free abelian group generated by isomorphism classes of objects modulo the relations $[B] = [A] + [C]$ for every short exact sequence $0 \to A \to B \to C \to 0$
• For a ring $R$, $K_0(R)$ is isomorphic to the Grothendieck group of the category of finitely generated projective $R$-modules
  • Elements of $K_0(R)$ represented by formal differences of isomorphism classes of projective modules
• $K_0(R)$ generalizes the notion of the ideal class group of a Dedekind domain, with the class group being isomorphic to the torsion subgroup of $K_0(R)$
• The rank map $K_0(R) \to \mathbb{Z}$ sends the class of a projective module to its rank, splitting $K_0(R)$ into a direct sum of the rank and the reduced $K_0$ group
• For a commutative ring $R$, $K_0(R)$ has a ring structure induced by the tensor product of modules, with the rank map being a ring homomorphism
• The Grothendieck-Riemann-Roch theorem relates the Chern character from K-theory to cohomology with the pushforward in K-theory and the Todd class
• $K_0(X)$ for a topological space $X$ is isomorphic to the Atiyah-Hirzebruch topological K-theory group $KU^0(X)$, defined using complex vector bundles on $X$

Higher K-Groups and Constructions

• Higher K-groups $K_n(R)$ of a ring $R$ are defined using the homotopy groups of the classifying space of the infinite general linear group $BGL(R)$ or its plus construction $BGL(R)^+$
  • $K_n(R) = \pi_n(BGL(R)^+)$ for $n \geq 1$
• The plus construction $BGL(R)^+$ is obtained from $BGL(R)$ by attaching cells to kill the maximal perfect subgroup of $\pi_1(BGL(R))$, resulting in a space with the same homology as $BGL(R)$ but with abelian fundamental group
• Quillen's Q-construction provides an alternative definition of higher K-groups using the category of bounded chain complexes of finitely generated projective $R$-modules
  • Homotopy groups of the geometric realization of the Q-construction agree with the K-groups defined using the plus construction
• The Quillen spectral sequence relates the higher K-groups of a ring to its Hochschild and cyclic homology
• The Milnor K-groups $K_n^M(F)$ of a field $F$ are defined using the tensor algebra of $F^\times$ modulo the Steinberg relations, with $K_n^M(F) = (F^\times)^{\otimes n} / \langle a_1 \otimes \cdots \otimes a_n : a_i + a_j = 1 \text{ for some } i \neq j \rangle$
• The Bloch-Kato conjecture (now a theorem due to Voevodsky) relates Milnor K-theory to étale cohomology and Galois cohomology
• The Beilinson regulator maps connect K-theory to Deligne cohomology and provide a bridge between K-theory and special values of L-functions

Applications in Topology and Algebra

• Topological K-theory (complex $KU^*$ and real $KO^*$) is a generalized cohomology theory that provides invariants of topological spaces and has applications in index theory and physics
  • Chern character connects topological K-theory to ordinary cohomology
• The Atiyah-Singer index theorem relates the analytic index of an elliptic operator on a manifold to the topological index defined using K-theory
• The K-theory of $C^*$-algebras (operator K-theory) has applications in noncommutative geometry and the classification of $C^*$-algebras
• The Baum-Connes conjecture relates the K-theory of the reduced group $C^*$-algebra of a group to the equivariant K-homology of its classifying space
• The Quillen-Lichtenbaum conjecture (now a theorem due to Voevodsky and Rost) relates the algebraic K-theory of a field to its étale cohomology, with applications in motivic homotopy theory
• The Quillen-Suslin theorem (formerly Serre's problem) states that every finitely generated projective module over a polynomial ring $k[x_1, \ldots, x_n]$ is free, with a proof using K-theory
• The K-theory of schemes has applications in intersection theory and the study of vector bundles on algebraic varieties
• The K-theory of categories (Waldhausen K-theory) has applications in the study of automorphisms of manifolds and the classification of high-dimensional manifolds

Connections to Other Mathematical Fields

• Algebraic K-theory is closely related to algebraic topology, with K-groups being homotopy groups of certain spaces or spectra
• The Quillen-Lichtenbaum conjecture connects algebraic K-theory to étale cohomology and motivic cohomology
• The Beilinson conjectures relate special values of L-functions to regulators in K-theory and motivic cohomology
• The Baum-Connes conjecture connects the K-theory of group $C^*$-algebras to the equivariant K-homology of classifying spaces
• Topological cyclic homology (TC) is a refinement of K-theory that has applications in the study of crystalline cohomology and p-adic Hodge theory
• The K-theory of derived categories and stable $\infty$-categories provides a framework for studying K-theory in the context of homological algebra and homotopy theory
• The K-theory of schemes is related to intersection theory and the study of characteristic classes in algebraic geometry
• The K-theory of number fields and their rings of integers has connections to number theory and the study of special values of zeta functions
• The K-theory of operator algebras (operator K-theory) is related to noncommutative geometry and the classification of $C^*$-algebras
• The K-theory of topological spaces is related to index theory and the study of elliptic operators on manifolds

Advanced Topics and Current Research

• Trace methods in K-theory, including the Dennis trace map and the cyclotomic trace, provide a connection between K-theory and cyclic homology theories
  • Used to study the K-theory of local fields and the étale K-theory of schemes
• Topological cyclic homology (TC) is a refinement of K-theory that captures p-adic information and has applications in p-adic Hodge theory and the study of crystalline cohomology
• The K-theory of derived categories and stable $\infty$-categories provides a framework for studying K-theory in the context of homological algebra and homotopy theory
  • Includes the K-theory of perfect complexes and the K-theory of dg-categories
• Motivic homotopy theory and motivic spectra provide a framework for studying K-theory in the context of algebraic geometry and relating it to motivic cohomology
• The study of the K-theory of ring spectra and structured ring spectra (E-infinity rings) is an active area of research, with applications in chromatic homotopy theory
• The relationship between K-theory and other invariants, such as topological Hochschild homology (THH) and topological André-Quillen homology (TAQ), is an active area of investigation
• The study of the K-theory of noncommutative spaces, such as quantum groups and noncommutative projective spaces, is a growing field with connections to noncommutative geometry
• The K-theory of singularities and the K-theory of quotient stacks are active areas of research with applications in equivariant K-theory and the study of orbifolds
• The development of computational methods for calculating K-groups and related invariants, such as the use of spectral sequences and the study of the K-theory of finite fields, is an ongoing area of research
• The application of K-theory to problems in mathematical physics, such as the classification of topological phases of matter and the study of D-brane charges in string theory, is a growing field of interdisciplinary research