Glossary

8.1 Basic theory of spectral sequences

3 min read•august 7, 2024

Spectral sequences are powerful tools in homological algebra, linking different levels of algebraic structures. They consist of pages of bigraded modules with differentials, where each page's homology determines the next page, ultimately converging to some limit.

Filtrations, exact couples, and edge homomorphisms are key concepts in understanding spectral sequences. These ideas help us analyze complex algebraic structures by breaking them down into simpler pieces and tracking how information flows between different levels of computation.

Spectral Sequences and Filtrations

Definition and Properties of Spectral Sequences

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  • Spectral sequence consists of a collection of pages {Erp,q}\{E_r^{p,q}\} indexed by non-negative integers rr called the page number
  • Each page is a bigraded module over a ring RR with a differential [dr](https://www.fiveableKeyTerm:dr)[d_r](https://www.fiveableKeyTerm:d_r) of bidegree (−r,r−1)(-r, r-1) satisfying dr∘dr=0d_r \circ d_r = 0
  • The pages are related by isomorphisms H(Er,dr)≅Er+1H(E_r, d_r) \cong E_{r+1} between the homology of the rr-th page and the (r+1)(r+1)-th page
  • Spectral sequences often arise from filtrations on a chain complex or from double complexes

Filtrations and Convergence

  • Filtration on a module MM is a sequence of submodules ⋯⊆Fp−1M⊆FpM⊆Fp+1M⊆⋯\cdots \subseteq F_{p-1}M \subseteq F_pM \subseteq F_{p+1}M \subseteq \cdots indexed by integers pp
  • Filtration gives rise to a spectral sequence by taking the associated graded module grpM=FpM/Fp−1M\mathrm{gr}_pM = F_pM/F_{p-1}M and defining the pages using the filtration
  • Spectral sequence is said to converge to a graded module H∗H_* if there exists a filtration on H∗H_* such that E∞p,q≅grpHp+qE_\infty^{p,q} \cong \mathrm{gr}_pH_{p+q} (E∞E_\infty page isomorphic to associated graded of the limit)
  • allows us to extract information about the limit module H∗H_* from the spectral sequence (E2E_2 and E∞E_\infty pages often of particular interest)

Exact and Derived Couples

Exact Couples and Spectral Sequences

  • Exact couple is a diagram of R-modules and homomorphisms A→iA→jE→kAA \xrightarrow{i} A \xrightarrow{j} E \xrightarrow{k} A where the composition of any two consecutive maps is zero
  • Exact couples give rise to spectral sequences by repeatedly taking homology to form derived couples
  • Derived couple of an exact couple (A,E,i,j,k)(A,E,i,j,k) is the exact couple (A′,E′,i′,j′,k′)(A',E',i',j',k') where A′=i(A)A' = i(A), E′=H(E)=ker⁡(k)/im(j)E' = H(E) = \ker(k)/\mathrm{im}(j), and the maps i′i', j′j', k′k' are induced by ii, jj, kk
  • Spectral sequence obtained from an exact couple has pages given by Er=E(r)E_r = E^{(r)} (the EE-module of the rr-th derived couple) and differentials induced by the map kk

Relationship between Exact Couples and Filtrations

  • Filtrations on a module MM give rise to exact couples by setting Ap=FpMA_p = F_pM and Ep=FpM/Fp−1ME_p = F_pM/F_{p-1}M with appropriate maps between them
  • Spectral sequence of this exact couple recovers the spectral sequence associated to the filtration
  • Exact couples provide a convenient way to construct and study spectral sequences, particularly in the context of filtered complexes and double complexes

Additional Concepts

Edge Homomorphisms

  • Edge homomorphisms in a spectral sequence are maps from the limit module H∗H_* to certain terms on the E2E_2 page or from certain E∞E_\infty terms to the limit module
  • For a first-quadrant spectral sequence {Erp,q}\{E_r^{p,q}\} converging to H∗H_*, there are edge homomorphisms Hn→E2n,0H_n \to E_2^{n,0} and E∞p,n−p→grpHnE_\infty^{p,n-p} \to \mathrm{gr}_pH_n for each nn and pp
  • Edge homomorphisms allow us to extract additional information about the limit module from the spectral sequence (E2E_2 and E∞E_\infty pages)
  • Example: In the Serre spectral sequence for a fibration F→E→BF \to E \to B, the edge homomorphisms relate the homology of the base BB and the fiber FF to the homology of the total space EE

Key Terms to Review (18)

Associated graded object: An associated graded object is a construction that arises from a filtered object, providing a way to study the filtration by examining its successive quotients. This concept is vital for analyzing the behavior of complexes and modules under filtration, allowing us to glean information about their structure and properties. By working with associated graded objects, one can simplify complex problems and gain insights into spectral sequences, which are tools used to compute homology and cohomology groups.
Cartan-Eilenberg Theorem: The Cartan-Eilenberg Theorem is a fundamental result in homological algebra that establishes a connection between the derived functors of a functor and the existence of a spectral sequence. It asserts that for a filtered complex, the associated spectral sequence converges to the homology of the total complex, providing a powerful tool for computing homological invariants.
Computing homology groups: Computing homology groups involves determining the algebraic invariants that characterize topological spaces, revealing their shape and structure. This process often uses tools like chain complexes and spectral sequences to manage complex calculations efficiently, especially in higher dimensions. The results provide insights into the features of spaces, such as holes and voids, which are fundamental in areas like algebraic topology.
Convergence: In the context of homological algebra, convergence refers to the process by which a spectral sequence approaches its limit, which represents a derived object or invariant. This concept is crucial because it determines how information from a filtered complex or a double complex can be systematically revealed and analyzed through the spectral sequence, ultimately leading to valuable topological or algebraic insights.
D_r: The term $d_r$ refers to the differential in the $r$-th page of a spectral sequence, which is a powerful tool in homological algebra that captures information about the algebraic structure of complexes. Each differential $d_r$ maps from one graded component of the spectral sequence to another, allowing for the computation of homology groups at each stage. The differentials play a crucial role in determining how the spectral sequence converges to the desired limit, often providing insight into the underlying topological or algebraic properties being studied.
Differential: In the context of algebraic topology and homological algebra, a differential is a linear map that connects two consecutive chain groups in a chain complex, typically denoted as d. It plays a crucial role in defining the structure of the complex, allowing one to analyze how elements in one degree relate to those in the next degree. Understanding differentials is essential for exploring various structures like spectral sequences and Koszul complexes, where they help in establishing relationships between different layers of algebraic data.
E-page: An e-page is a specific page in the context of spectral sequences, often denoted as E_r, where r indicates the page number. This concept is crucial for analyzing how spectral sequences converge and the structures that arise from them, providing a systematic way to compute derived functors or cohomology groups. The e-pages are constructed using differentials that allow us to track how information changes as we move through the spectral sequence.
E^{p,q}_r: The term $e^{p,q}_r$ refers to the $r$-th page of a spectral sequence that arises from a double complex, specifically capturing the differentials and the relationships between the homology groups at various filtration levels. This notation is crucial as it helps organize the information in spectral sequences, allowing for computations of homological invariants and understanding convergence properties. Essentially, it serves as a structured way to analyze how complex structures can simplify through successive approximations.
Filtering: Filtering is a concept in mathematics that involves a way of organizing a set or a structure into smaller, more manageable pieces, known as filters. This method allows for the analysis of structures by breaking them down into layers or stages, making it easier to study properties and relationships within the larger context. In the realm of spectral sequences, filtering is crucial as it helps in controlling convergence and allows for a clearer understanding of how complex algebraic structures can be approximated.
First Page of a Spectral Sequence: The first page of a spectral sequence refers to the initial set of abutments and differentials that provide a foundational layer for calculating homological invariants. This page is critical because it captures the essential information that will evolve through subsequent pages, enabling computations in algebraic topology and derived categories. The entries on this page are typically derived from a filtered complex or a double complex, setting the stage for the spectral sequence's convergence and eventual outcome.
Henri Cartan: Henri Cartan was a prominent French mathematician known for his significant contributions to algebraic topology and homological algebra, particularly in developing the theory of sheaves and derived functors. His work laid essential groundwork for later developments in category theory and cohomology, impacting various mathematical areas including group cohomology and spectral sequences.
Jean-Pierre Serre: Jean-Pierre Serre is a prominent French mathematician known for his work in algebraic geometry, topology, and number theory. His contributions have had a significant impact on various areas of mathematics, particularly through the development of cohomological methods and the theory of derived categories.
Leray Spectral Sequence: The Leray spectral sequence is a powerful tool in algebraic topology and homological algebra that provides a way to compute homology groups of a topological space by analyzing the structure of a fibration. It connects the homology of a total space, base space, and fiber, effectively allowing one to understand complex spaces through simpler ones. This concept is crucial for working with filtered complexes, double complexes, and has important applications in deriving significant results in homological algebra.
Spectral sequence associated to a filtration: A spectral sequence associated to a filtration is a computational tool in homological algebra that arises from a filtered complex or a filtered chain. It allows one to compute homology groups by systematically organizing and collapsing data through successive approximations, ultimately providing insights into the structure of the underlying object. This concept is fundamental in understanding how filtered complexes relate to their associated homological invariants and how these invariants behave under certain operations.
Spectral sequence for sheaf cohomology: A spectral sequence for sheaf cohomology is a computational tool that helps to derive sheaf cohomology groups from a filtered complex of sheaves. It provides a systematic way to compute the cohomology of a space by organizing data into a sequence of pages, each consisting of groups and differentials, leading to the final cohomology groups. This method is particularly useful when dealing with complex spaces or sheaves where direct computation is difficult.
Spectral sequence of a filtered complex: A spectral sequence of a filtered complex is a mathematical tool used in homological algebra to analyze the properties of a filtered chain complex. It arises from the filtration on the complex and provides a way to compute derived functors, allowing one to systematically track how different components interact through successive approximations. This method helps in extracting important information about homology groups while simplifying complex calculations.
Theorem on Convergence of Spectral Sequences: The theorem on convergence of spectral sequences provides criteria under which a spectral sequence converges to a certain type of limit, often related to derived functors or homology groups. This theorem plays a crucial role in understanding how spectral sequences can be used to compute homological algebra objects by establishing when the terms of the sequence stabilize and yield meaningful results.
Truncation: Truncation is the process of cutting off or limiting the elements of a mathematical structure, often to make it more manageable or focused. In the context of spectral sequences, truncation can be particularly useful for simplifying complex data, enabling the extraction of relevant information while ignoring higher-level intricacies that may not be necessary for immediate analysis.
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