8.1 Basic theory of spectral sequences

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

Filtrations and Convergence

• Filtration on a module $M$ is a sequence of submodules $\cdots \subseteq F_{p-1}M \subseteq F_pM \subseteq F_{p+1}M \subseteq \cdots$ indexed by integers $p$
• Filtration gives rise to a spectral sequence by taking the associated graded module $\mathrm{gr}_pM = F_pM/F_{p-1}M$ and defining the pages using the filtration
• Spectral sequence is said to converge to a graded module $H_*$ if there exists a filtration on $H_*$ such that $E_\infty^{p,q} \cong \mathrm{gr}_pH_{p+q}$ ($E_\infty$ page isomorphic to associated graded of the limit)
• Convergence allows us to extract information about the limit module $H_*$ from the spectral sequence ($E_2$ and $E_\infty$ pages often of particular interest)

Exact and Derived Couples

Exact Couples and Spectral Sequences

• Exact couple is a diagram of R-modules and homomorphisms $A \xrightarrow{i} A \xrightarrow{j} E \xrightarrow{k} A$ 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 $(A,E,i,j,k)$ is the exact couple $(A',E',i',j',k')$ where $A' = i(A)$, $E' = H(E) = \ker(k)/\mathrm{im}(j)$, and the maps $i'$, $j'$, $k'$ are induced by $i$, $j$, $k$
• Spectral sequence obtained from an exact couple has pages given by $E_r = E^{(r)}$ (the $E$-module of the $r$-th derived couple) and differentials induced by the map $k$

Relationship between Exact Couples and Filtrations

• Filtrations on a module $M$ give rise to exact couples by setting $A_p = F_pM$ and $E_p = F_pM/F_{p-1}M$ 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 $H_*$ to certain terms on the $E_2$ page or from certain $E_\infty$ terms to the limit module
• For a first-quadrant spectral sequence $\{E_r^{p,q}\}$ converging to $H_*$, there are edge homomorphisms $H_n \to E_2^{n,0}$ and $E_\infty^{p,n-p} \to \mathrm{gr}_pH_n$ for each $n$ and $p$
• Edge homomorphisms allow us to extract additional information about the limit module from the spectral sequence ($E_2$ and $E_\infty$ pages)
• Example: In the Serre spectral sequence for a fibration $F \to E \to B$, the edge homomorphisms relate the homology of the base $B$ and the fiber $F$ to the homology of the total space $E$