🔢Algebraic K-Theory Unit 5 – Spectral Sequences: Atiyah-Hirzebruch Method

Key Concepts and Definitions

• Spectral sequence: Algebraic tool used to compute homology or cohomology groups by successively approximating the desired groups
• Atiyah-Hirzebruch spectral sequence (AHSS): Specific type of spectral sequence used to calculate generalized cohomology theories, such as K-theory, from ordinary cohomology
• Filtration: Sequence of subspaces or subgroups used to construct the spectral sequence
  • Each filtration level corresponds to a page in the spectral sequence
• Differentials: Maps between the terms on each page of the spectral sequence that provide successive approximations
• Convergence: Property of the spectral sequence where the terms stabilize after a finite number of pages, yielding the desired cohomology groups
• Associated graded: Graded object constructed from a filtered object, used in the construction of the AHSS
• Exact couple: Algebraic structure consisting of two groups and three homomorphisms, used to define the differentials in the spectral sequence

Historical Context and Development

• Spectral sequences first introduced by Jean Leray in the 1940s to study the cohomology of fiber bundles
• Atiyah and Hirzebruch adapted the concept to generalized cohomology theories in the 1960s, leading to the development of the AHSS
• The AHSS provided a powerful tool for computing K-theory and other cohomology theories in terms of ordinary cohomology
• Serre's work on the cohomology of fiber bundles and the Serre spectral sequence influenced the development of the AHSS
• Grothendieck's work on algebraic geometry and the theory of schemes also contributed to the understanding and applications of spectral sequences
• The AHSS has since been widely used in algebraic topology, algebraic geometry, and other areas of mathematics

Atiyah-Hirzebruch Spectral Sequence Basics

• The AHSS is a spectral sequence that relates a generalized cohomology theory $h^*(X)$ of a topological space $X$ to its ordinary cohomology $H^*(X)$
• Constructed using a filtration of the space $X$ by its skeleta $X_0 \subset X_1 \subset \cdots \subset X$
  • The filtration induces a filtration on the generalized cohomology $h^*(X)$
• The $E_2$ page of the AHSS is given by $E_2^{p,q} = H^p(X; h^q(pt))$, where $h^q(pt)$ is the generalized cohomology of a point
• The differentials on the $E_r$ page have bidegree $(r, 1-r)$ and are determined by the generalized cohomology theory $h^*$
• The AHSS converges to the associated graded of the filtered generalized cohomology $h^*(X)$
  • Convergence is denoted by $E_2^{p,q} \Rightarrow h^{p+q}(X)$

Construction and Properties

• The AHSS is constructed using the skeletal filtration of the space $X$, where $X_n$ is the $n$-skeleton of $X$
• The filtration induces a long exact sequence of generalized cohomology groups $\cdots \to h^*(X_n) \to h^*(X_{n+1}) \to h^*(X_{n+1}, X_n) \to \cdots$
• The relative generalized cohomology groups $h^*(X_{n+1}, X_n)$ can be identified with the ordinary cohomology of $X$ with coefficients in $h^*(pt)$
  • This identification uses the suspension isomorphism and the fact that $(X_{n+1}, X_n)$ is homotopy equivalent to a wedge of $n$-spheres
• The long exact sequences for different values of $n$ form an exact couple, which gives rise to the spectral sequence
• The differentials in the AHSS are determined by the connecting homomorphisms in the long exact sequences and the multiplication in the generalized cohomology theory
• The AHSS is natural with respect to maps of spaces and generalized cohomology theories
  • A map $f: X \to Y$ induces a map of spectral sequences $f^*: E_r^{*,*}(Y) \to E_r^{*,*}(X)$

Applications in Algebraic K-Theory

• The AHSS is a powerful tool for computing the algebraic K-theory of a scheme or a ring
• For a scheme $X$, the AHSS relates the algebraic K-theory $K_*(X)$ to the motivic cohomology $H^{*,*}(X, \mathbb{Z})$
  • The $E_2$ page is given by $E_2^{p,q} = H^{p,q}(X, \mathbb{Z})$, where $H^{p,q}(X, \mathbb{Z})$ is the motivic cohomology of $X$
• For a ring $R$, the AHSS relates the algebraic K-theory $K_*(R)$ to the Hochschild homology $HH_*(R)$ and the cyclic homology $HC_*(R)$
  • The $E_2$ page involves the Hochschild and cyclic homology of $R$
• The AHSS has been used to prove important results in algebraic K-theory, such as the Quillen-Lichtenbaum conjecture
• The Atiyah-Hirzebruch spectral sequence can also be used to study the relationship between algebraic K-theory and other invariants, such as the Chow groups and the étale cohomology

Computational Techniques and Examples

• The AHSS can be used to compute the generalized cohomology of a space by successively approximating the cohomology groups using the differentials
• To compute the differentials, one often uses the structure of the generalized cohomology theory and the geometry of the space
• For example, in the case of K-theory, the differentials are related to the Steenrod operations and the Chern character
• Computational techniques include:
  • The use of the Serre spectral sequence to compute the ordinary cohomology of the space
  • The use of the Adams spectral sequence to compute the generalized cohomology of a point
  • The use of the Bockstein spectral sequence to study the torsion in the cohomology groups
• Examples of computations using the AHSS:
  • Computing the complex K-theory of projective spaces and Grassmannians
  • Computing the algebraic K-theory of fields and rings of integers
  • Computing the topological K-theory of classifying spaces of compact Lie groups

Related Theories and Extensions

• The AHSS is closely related to other spectral sequences in algebraic topology and algebraic geometry, such as the Serre spectral sequence and the Eilenberg-Moore spectral sequence
• The motivic Atiyah-Hirzebruch spectral sequence is an extension of the AHSS to the setting of motivic homotopy theory
  • It relates the motivic generalized cohomology theories to the motivic ordinary cohomology
• The equivariant Atiyah-Hirzebruch spectral sequence is an extension of the AHSS to the setting of equivariant topology
  • It relates the equivariant generalized cohomology theories to the equivariant ordinary cohomology
• The Atiyah-Hirzebruch spectral sequence has also been generalized to the setting of triangulated categories and derived categories
  • These generalizations have applications in derived algebraic geometry and the study of motives

Challenges and Open Problems

• Computing the differentials in the AHSS can be a challenging problem, especially for higher pages of the spectral sequence
  • Requires a deep understanding of the generalized cohomology theory and the geometry of the space
• Convergence of the AHSS is not always guaranteed, and there are examples of non-convergent spectral sequences
  • Understanding the convergence properties of the AHSS in various settings is an active area of research
• Extending the AHSS to more general settings, such as non-commutative spaces or derived schemes, is an ongoing challenge
• Applying the AHSS to compute the generalized cohomology of interesting spaces or schemes, such as the moduli spaces of curves or the classifying spaces of algebraic groups
• Understanding the relationship between the AHSS and other spectral sequences, such as the descent spectral sequence or the Hodge-to-de Rham spectral sequence
• Using the AHSS to study the relationship between different generalized cohomology theories, such as K-theory and cobordism theory