🧬Homological Algebra Unit 8 – Spectral Sequences

What's the Big Idea?

• Spectral sequences provide a powerful computational tool in homological algebra to calculate homology groups of complex objects
• Break down complex calculations into a sequence of simpler steps, allowing for the gradual approximation of the desired homology groups
• Arise naturally in various contexts, such as the study of fibrations, group cohomology, and the Leray-Serre spectral sequence
• Enable the systematic study of the relationships between different homology theories and their associated long exact sequences
• Provide a framework for organizing and simplifying complex algebraic structures and computations
• Allow for the derivation of important results and invariants in algebraic topology and geometry
• Serve as a bridge between different branches of mathematics, connecting ideas from algebra, topology, and geometry

Key Concepts and Definitions

• Spectral sequence: a collection of differential bigraded modules $\{E_r^{p,q}\}$ with differentials $d_r: E_r^{p,q} \to E_r^{p+r,q-r+1}$ satisfying $d_r \circ d_r = 0$
• Convergence: a spectral sequence is said to converge to a graded module $H^*$ if there exists an $r_0$ such that $E_{r_0}^{p,q} \cong E_\infty^{p,q}$ for all $p,q$, and $H^n \cong \bigoplus_{p+q=n} E_\infty^{p,q}$
• Filtration: a decreasing sequence of submodules $\cdots \subseteq F^{p+1}M \subseteq F^pM \subseteq \cdots$ of a module $M$
  • Spectral sequences often arise from filtrations on chain complexes or modules
• Differential bigraded module: a bigraded module $E = \bigoplus_{p,q} E^{p,q}$ equipped with a differential $d: E^{p,q} \to E^{p',q'}$ satisfying $d \circ d = 0$
• Exact couple: a pair of bigraded modules $(D,E)$ with maps $i: D \to D$, $j: D \to E$, and $k: E \to D$ satisfying certain axioms
  • Exact couples give rise to spectral sequences via the derived couple construction
• Convergence to the associated graded: a spectral sequence $\{E_r^{p,q}\}$ converges to the associated graded of a filtered module $M$ if $E_\infty^{p,q} \cong F^pM/F^{p+1}M$

Types of Spectral Sequences

• Leray-Serre spectral sequence: relates the homology of a fibration to the homology of its base and fiber
  • Arises from the filtration of the total space by the preimages of the skeleta of the base
• Grothendieck spectral sequence: relates the derived functors of a composite functor to the derived functors of its components
• Lyndon-Hochschild-Serre spectral sequence: relates the group cohomology of a group extension to the cohomology of its factors
• Adams spectral sequence: computes stable homotopy groups of spheres using the Adams filtration on the stable homotopy category
• Eilenberg-Moore spectral sequence: relates the homology of a pullback to the homology of its components
• Bockstein spectral sequence: arises from the filtration of a chain complex by the powers of an ideal in the coefficient ring
• Hypercohomology spectral sequence: computes the hypercohomology of a complex of sheaves using a suitable resolution

How Spectral Sequences Work

• Start with a filtered complex or a double complex, which gives rise to a spectral sequence
• The spectral sequence consists of a collection of pages $\{E_r^{p,q}\}$, each with a differential $d_r: E_r^{p,q} \to E_r^{p+r,q-r+1}$
• The pages are related by the formula $E_{r+1}^{p,q} \cong H(E_r^{p,q}, d_r)$, i.e., each page is the homology of the previous page with respect to its differential
• The differentials satisfy the property $d_r \circ d_r = 0$, ensuring that the homology is well-defined
• As $r$ increases, the pages stabilize, meaning that for sufficiently large $r$, $E_r^{p,q} \cong E_{r+1}^{p,q}$ for all $p,q$
• The stable page, denoted $E_\infty^{p,q}$, is the limit of the spectral sequence and often encodes important information about the original complex or the associated graded of a filtered module
• Convergence of the spectral sequence to a graded module $H^*$ means that $H^n$ can be reconstructed from the stable page $E_\infty^{p,q}$ with $p+q=n$

Applications in Homological Algebra

• Computing the homology and cohomology of chain complexes and topological spaces
  • Spectral sequences can simplify calculations by breaking them down into smaller, more manageable steps
• Studying the relationship between different homology theories, such as singular homology and Čech cohomology
• Calculating the homology of group extensions and fibrations
  • The Lyndon-Hochschild-Serre and Leray-Serre spectral sequences are particularly useful in these contexts
• Investigating the structure of derived functors and their compositions
  • The Grothendieck spectral sequence is a powerful tool for understanding the behavior of derived functors
• Computing the hypercohomology of complexes of sheaves on topological spaces or schemes
• Analyzing the Adams filtration on the stable homotopy category and computing stable homotopy groups of spheres
• Studying the homological properties of rings and modules, such as the Bockstein homomorphism and the structure of Ext and Tor functors

Computational Techniques

• Identify the appropriate spectral sequence for the problem at hand, based on the available data and the desired output
• Set up the initial page of the spectral sequence, usually denoted $E_1^{p,q}$ or $E_2^{p,q}$, using the given data (e.g., a filtration or a double complex)
• Compute the differentials on each page using the formula $d_r: E_r^{p,q} \to E_r^{p+r,q-r+1}$ and the properties of the specific spectral sequence
• Calculate the homology of each page with respect to its differential, i.e., $E_{r+1}^{p,q} \cong H(E_r^{p,q}, d_r)$
• Continue the process until the pages stabilize, i.e., $E_r^{p,q} \cong E_{r+1}^{p,q}$ for all $p,q$ and sufficiently large $r$
• Interpret the stable page $E_\infty^{p,q}$ in terms of the original problem, such as the homology of a complex or the associated graded of a filtered module
• Use the convergence properties of the spectral sequence to reconstruct the desired output, such as the homology groups $H^n$, from the stable page $E_\infty^{p,q}$

Common Pitfalls and How to Avoid Them

• Incorrectly identifying the appropriate spectral sequence for a given problem
  • Carefully analyze the available data and the desired output to choose the most suitable spectral sequence
• Miscomputing the differentials or the homology of each page
  • Double-check the formulas and properties of the specific spectral sequence being used, and verify that the differentials satisfy $d_r \circ d_r = 0$
• Failing to recognize when the spectral sequence has stabilized
  • Keep track of the changes between consecutive pages and look for patterns that indicate stabilization
• Misinterpreting the stable page or the convergence properties of the spectral sequence
  • Pay close attention to the grading and the structure of the stable page, and understand how it relates to the original problem
• Neglecting the possibility of extension problems when reconstructing the output from the stable page
  • Be aware that the stable page may not always provide complete information about the desired output, and additional work may be needed to resolve extension issues
• Overcomplicating the calculation by using a spectral sequence when a simpler method would suffice
  • Consider alternative approaches and weigh the benefits of using a spectral sequence against the potential complexity of the computation

Real-World Examples and Cool Stuff

• The Leray-Serre spectral sequence can be used to compute the cohomology of the loop space of a topological group, which has applications in mathematical physics and string theory
• The Adams spectral sequence is a powerful tool for studying the stable homotopy groups of spheres, which are central objects in algebraic topology and have connections to cobordism theory and K-theory
• Spectral sequences have been used to prove the Weil conjectures, which relate the topology of algebraic varieties over finite fields to their arithmetic properties
• The Eilenberg-Moore spectral sequence has applications in the study of group actions on topological spaces and the equivariant cohomology of transformation groups
• Spectral sequences have been employed in the computation of the K-theory of various algebraic objects, such as rings, schemes, and categories
• The Bockstein spectral sequence is useful for analyzing the torsion in the homology of topological spaces and has connections to the study of Steenrod squares and cohomology operations
• Spectral sequences have played a crucial role in the development of modern algebraic geometry, particularly in the study of sheaf cohomology and the theory of motives