🔢Algebraic K-Theory Unit 1 – 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
• K-groups $K_i(R)$ are abelian groups associated to a ring $R$ that capture arithmetic and geometric properties
• Grothendieck group $K_0(R)$ is the zeroth K-group, constructed from finitely generated projective modules over $R$
• Higher K-groups $K_i(R)$ for $i > 0$ are defined using Quillen's Q-construction or Waldhausen's S-construction
  • $K_1(R)$ is closely related to the group of units $GL(R)$ and captures information about determinants and commutators
  • $K_2(R)$ is related to the Steinberg group and universal central extensions
• Negative K-groups $K_{-i}(R)$ are defined using Bass's Nil groups and capture nilpotence in the ring
• Milnor K-theory $K_i^M(F)$ is a graded ring associated to a field $F$, related to the norm residue symbol and motivic cohomology

Historical Context and Development

• Algebraic K-theory originated in the work of Alexander Grothendieck in the 1950s on the Riemann-Roch theorem and the Grothendieck group
• Hyman Bass developed algebraic K-theory for rings in the 1960s, introducing negative K-groups and the Bass-Heller-Swan decomposition
• Daniel Quillen revolutionized the field in the 1970s with his higher K-theory and the Q-construction, enabling the use of homotopy-theoretic methods
  • Quillen established fundamental properties like long exact sequences, localization, and devissage
• Friedhelm Waldhausen introduced the S-construction in the 1980s, providing an alternative approach to higher K-theory
• Algebraic K-theory has since found connections to various areas of mathematics, including algebraic geometry, number theory, and topology
  • Notably, the Quillen-Lichtenbaum conjecture relates algebraic K-theory to étale cohomology and the Bloch-Kato conjecture

Fundamental Groups and Categories

• Fundamental groups in topology, such as $\pi_1(X)$ for a topological space $X$, capture information about loops and homotopy classes
• In algebraic K-theory, the fundamental groupoid $\Pi_1(\mathcal{C})$ of a category $\mathcal{C}$ plays a similar role
  • Objects of $\Pi_1(\mathcal{C})$ are the objects of $\mathcal{C}$, and morphisms are homotopy classes of paths
• The Q-construction $Q(\mathcal{C})$ is a simplicial set associated to a category $\mathcal{C}$, whose geometric realization is the K-theory space
  • $\pi_i(Q(\mathcal{C})) \cong K_i(\mathcal{C})$ relates the homotopy groups of the Q-construction to the K-groups
• The S-construction $S_\bullet(\mathcal{C})$ is a simplicial category that also yields the K-theory space upon geometric realization
• Fundamental theorems like the Additivity Theorem and the Resolution Theorem govern the behavior of K-theory under certain categorical constructions

K0 and Grothendieck Groups

• The Grothendieck group $K_0(\mathcal{C})$ of an abelian category $\mathcal{C}$ is the free abelian group generated by isomorphism classes $[X]$ of objects, modulo the relation $[X] = [Y] + [Z]$ for short exact sequences $0 \to Y \to X \to Z \to 0$
  • For the category of finitely generated projective modules over a ring $R$, this yields the zeroth K-group $K_0(R)$
• $K_0(R)$ can be computed using the idempotent completion and the split exact sequences
  • Idempotents $e \in M_n(R)$ give rise to finitely generated projective modules $eR^n$
• The rank map $K_0(R) \to \mathbb{Z}$ sends the class of a projective module to its rank, yielding the augmented K-theory $\tilde{K}_0(R) = \ker(K_0(R) \to \mathbb{Z})$
• For a Noetherian scheme $X$, the Grothendieck group $K_0(X)$ is generated by coherent sheaves, with relations from exact sequences
  • $K_0(X)$ is a ring under tensor product, and the Euler characteristic $\chi : K_0(X) \to \mathbb{Z}$ is a ring homomorphism

Higher K-Groups and Constructions

• Higher K-groups $K_i(R)$ for $i > 0$ capture more subtle arithmetic and geometric information about the ring $R$
• Quillen's Q-construction $Q(R)$ is a simplicial set whose homotopy groups yield the higher K-groups: $\pi_i(Q(R)) \cong K_i(R)$
  • The Q-construction is defined using the category of finitely generated projective $R$-modules and their automorphisms
• Waldhausen's S-construction $S_\bullet(R)$ is a simplicial category that also yields the higher K-groups upon taking homotopy groups of its geometric realization
  • The S-construction is more general and can be applied to categories with cofibrations and weak equivalences
• The plus construction $BGL(R)^+$ is a space obtained by attaching cells to the classifying space $BGL(R)$ to kill the perfect normal subgroups
  • $\pi_i(BGL(R)^+) \cong K_i(R)$ for $i \geq 1$, providing a more geometric interpretation of the higher K-groups
• The Milnor K-groups $K_i^M(F)$ of a field $F$ are defined using tensor products and quotients of the multiplicative group $F^\times$
  • Milnor K-theory is related to algebraic K-theory via the norm residue homomorphism $K_i^M(F) \to K_i(F)$

Applications in Algebra and Topology

• Algebraic K-theory has numerous applications in various branches of mathematics, connecting arithmetic, geometry, and topology
• In algebraic geometry, K-theory is used to study vector bundles, coherent sheaves, and the Riemann-Roch theorem
  • The Grothendieck-Riemann-Roch theorem expresses the Euler characteristic in terms of Chern characters and Todd classes
• In number theory, K-theory is related to the study of algebraic number fields, class groups, and the Brauer group
  • The Quillen-Lichtenbaum conjecture relates the K-theory of rings of integers to étale cohomology and special values of L-functions
• In topology, K-theory is connected to the study of vector bundles, characteristic classes, and generalized cohomology theories
  • The Bott periodicity theorem establishes a periodic behavior in the K-theory of topological spaces
• K-theory also has applications in operator algebras, where it is used to study projections, unitaries, and the classification of C*-algebras
  • The Baum-Connes conjecture relates the K-theory of group C*-algebras to the geometry and topology of the group

Computational Techniques and Examples

• Computing K-groups is a challenging task, but various techniques and tools have been developed to aid in calculations
• For the zeroth K-group $K_0(R)$, the idempotent completion and split exact sequences provide a means of computation
  • Example: For the ring of integers $\mathbb{Z}$, $K_0(\mathbb{Z}) \cong \mathbb{Z}$, generated by the class of the free module $\mathbb{Z}$
• The Fundamental Theorem of K-theory relates the K-theory of a ring $R$ to that of the polynomial ring $R[t]$ and the Laurent polynomial ring $R[t, t^{-1}]$
  • This allows for inductive computations and the use of the Bass-Heller-Swan decomposition
• Spectral sequences, such as the Atiyah-Hirzebruch spectral sequence and the Quillen spectral sequence, can be used to compute K-groups
  • These spectral sequences relate K-theory to other cohomology theories and provide a means of computation via filtrations and exact couples
• For specific classes of rings or schemes, such as regular rings or smooth varieties, there are often more explicit formulas or techniques available
  • Example: For a smooth projective curve $X$ over a field, $K_0(X) \cong \mathbb{Z} \oplus \text{Pic}(X)$, where $\text{Pic}(X)$ is the Picard group of line bundles on $X$

Advanced Topics and Current Research

• Algebraic K-theory continues to be an active area of research, with numerous advances and open problems
• Motivic homotopy theory and motivic cohomology provide a framework for studying algebraic varieties and schemes from a homotopical perspective
  • The motivic Bloch-Kato conjecture relates Milnor K-theory to étale cohomology and has important consequences for the structure of K-groups
• Trace methods, such as the Dennis trace and the cyclotomic trace, relate K-theory to other invariants like Hochschild homology and topological cyclic homology
  • The Lichtenbaum-Quillen conjecture, proved by Voevodsky, states that the motivic cohomology of a field agrees with its étale cohomology after inverting a certain integer
• Equivariant K-theory studies the K-theory of schemes or spaces equipped with a group action, leading to interesting representation-theoretic aspects
  • The Baum-Connes conjecture, relating the equivariant K-theory of the classifying space to the K-theory of the reduced group C*-algebra, is a major open problem
• K-theory of derived categories and triangulated categories has become increasingly important, with connections to representation theory and algebraic geometry
  • The Thomason-Trobaugh theorem establishes a localization sequence for the K-theory of derived categories of perfect complexes
• Other active areas of research include the K-theory of singularities, the K-theory of noncommutative rings and algebras, and the relationship between K-theory and other invariants like cyclic homology and topological Hochschild homology