8.2 Spectral sequence of a filtered complex

Spectral sequences are powerful tools for computing homology groups of complex algebraic structures. They work by filtering a complex and analyzing the resulting graded objects, providing a systematic approach to unraveling complicated homological information.

This section focuses on the spectral sequence of a filtered complex, introducing key concepts like filtrations, graded objects, and spectral sequence pages. We'll explore how these ideas come together to create a powerful computational framework for homological algebra.

Filtered Complexes and Graded Objects

Filtered Complexes

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• Filtered complex consists of a chain complex $C$ and a family of subcomplexes $F^pC$ indexed by integers $p$
  • Subcomplexes satisfy $F^pC \subseteq F^{p+1}C$ for all $p$
  • Entire complex is the union of all subcomplexes $C = \bigcup_p F^pC$
• Filtration gives a notion of "size" or "degree" to elements of the complex
  • Elements in $F^pC$ are considered to have filtration degree at least $p$
• Morphisms between filtered complexes $f: (C, F) \to (D, G)$ preserve the filtration
  • Require $f(F^pC) \subseteq G^pD$ for all $p$

Graded Modules and Associated Graded Objects

• Graded module $M$ is a direct sum of submodules $M = \bigoplus_{n \in \mathbb{Z}} M_n$
  • Elements in $M_n$ are homogeneous of degree $n$
  • Homomorphisms between graded modules preserve the grading
• Associated graded object $\operatorname{gr}(C, F)$ of a filtered complex $(C, F)$ is a graded module
  • Defined as $\operatorname{gr}(C, F)_p = F^pC / F^{p-1}C$
  • Measures the "jump" in filtration degree from $p-1$ to $p$
• Associated graded object captures the structure of the filtration
  • Loses some information about the complex $C$ itself
  • Useful for studying properties that depend on the filtration ($\operatorname{gr}$ functor)

Spectral Sequence Pages

The $E_0$ and $E_1$ Pages

• $E_0$ page of a spectral sequence is the associated graded object of the filtered complex
  • $E_0^{p,q} = \operatorname{gr}(C, F)_p^q = F^pC^q / F^{p-1}C^q$
  • Differential $d_0: E_0^{p,q} \to E_0^{p,q-1}$ induced by the differential of $C$
• $E_1$ page is the homology of the $E_0$ page with respect to $d_0$
  • $E_1^{p,q} = H^q(E_0^{p,*}, d_0) = \ker(d_0: E_0^{p,q} \to E_0^{p,q-1}) / \operatorname{im}(d_0: E_0^{p,q+1} \to E_0^{p,q})$
  • Differential $d_1: E_1^{p,q} \to E_1^{p-1,q}$ induced by $d_0$ (zigzag homomorphism)

The Higher Pages and Differentials

• $E_r$ page for $r \geq 1$ is obtained by taking homology of the previous page $E_{r-1}$
  • $E_r^{p,q} = H^q(E_{r-1}^{p,*}, d_{r-1}) = \ker(d_{r-1}: E_{r-1}^{p,q} \to E_{r-1}^{p-r+1,q+r-2}) / \operatorname{im}(d_{r-1}: E_{r-1}^{p+r-1,q-r+2} \to E_{r-1}^{p,q})$
  • Differential $d_r: E_r^{p,q} \to E_r^{p-r,q+r-1}$ has bidegree $(-r, r-1)$
• Spectral sequence pages form a sequence of successive approximations to the homology of the original complex $C$
  • Each page $E_r$ is a bigraded module with a differential $d_r$
  • Homology of $(E_r, d_r)$ gives the next page $E_{r+1}$

Convergence and Degeneration

Convergence of Spectral Sequences

• Spectral sequence $\{E_r^{p,q}, d_r\}$ of a filtered complex $(C, F)$ is said to converge to $H^*(C)$ if there exists an $r_0$ such that for all $r \geq r_0$:
  • $E_r^{p,q} \cong E_{r_0}^{p,q}$ (isomorphic as bigraded modules)
  • $d_r = 0$ (differentials vanish)
• Convergence theorem states that under certain conditions (bounded or exhaustive filtration), the spectral sequence converges to the associated graded object of $H^*(C)$ with respect to the induced filtration
  • $E_\infty^{p,q} \cong \operatorname{gr}(H^*(C), F)_p^q = F^pH^{p+q}(C) / F^{p-1}H^{p+q}(C)$
• Convergence allows us to compute the homology of the original complex $C$ from the limit term $E_\infty$ of the spectral sequence
  • Requires knowledge of the induced filtration on $H^*(C)$

Degeneration and Collapsing

• Spectral sequence is said to degenerate at the $E_r$ page if $d_r = 0$ and all subsequent differentials also vanish
  • Implies $E_r^{p,q} \cong E_{r+1}^{p,q} \cong \cdots \cong E_\infty^{p,q}$ for all $p, q$
• Spectral sequence collapses at the $E_r$ page if it degenerates at $E_r$ and $E_r = E_\infty$
  • Stronger condition than degeneration
  • Collapsing at $E_1$ or $E_2$ is particularly useful for computations
• Degeneration and collapsing simplify the calculation of the limit term $E_\infty$
  • Avoid the need to compute higher differentials
  • Provide a direct relationship between the pages and the homology of the original complex