Approximate the complex, then correct
A filtration of a chain complex breaks the differential into pieces according to how far they move through the filtration. The associated graded complex uses only the part that preserves the filtration degree, and its homology is the first approximation. Successive pages account for the parts that move one step, two steps, and so on. That is the entire conceptual content of a spectral sequence of a filtered complex.
Learning objectives
- Define a filtration and the associated graded complex.
- Explain how the pages successively correct the approximation.
- Distinguish bounded, exhaustive and complete filtrations.
- Identify the filtration on the target that E∞ describes.
Section 01Filtrations
An increasing filtration is a chain of subcomplexes … ⊆ Fp−1C ⊆ FpC ⊆ … with associated graded
| Condition | Meaning | Why it matters |
|---|---|---|
| Bounded | FpC = 0 for small p and = C for large p, in each degree | Guarantees convergence with no further hypotheses |
| Exhaustive | The union of all FpC is C | Necessary for the sequence to see all of C |
| Hausdorff | The intersection of all FpC is 0 | Prevents information being invisible to the filtration |
| Complete | C is the inverse limit of C/FpC | Needed for convergence in the unbounded case, along with vanishing lim1 |
Section 02The pages as successive corrections
- Stage 01E0The associated graded complex itself, with only the filtration-preserving part of the differential.
- Stage 02E1Its homology — the first approximation to H(C), correct if the differential never left the filtration degree.
- Stage 03E2Corrects for the part of the differential that drops filtration by one step.
- Stage 04ErCorrects for drops of r−1 steps. Eventually every correction has been applied.
- Stage 05E∞The associated graded of the induced filtration on H(C) — not H(C) itself.
A spectral sequence computes the homology of the associated graded and then corrects it, page by page, for everything the grading discarded. The corrections are the differentials, and E∞ is what remains.
Section 03The extension problem
E∞p,q is FpHn / Fp−1Hn where n = p + q. Reconstructing Hn requires solving a chain of extension problems.
If E∞ in total degree 1 has pieces ℤ/2 and ℤ/2, the target could be ℤ/2 ⊕ ℤ/2 or ℤ/4. The spectral sequence does not distinguish them. Additional input — a ring structure, a naturality argument, a known special case — is required.
Over a field the problem disappears, since every extension of vector spaces splits. This is one more reason field coefficients are preferred when the extension data is not needed.
ReferenceFrequently asked questions
Which direction should the filtration run?
Either, with the appropriate conventions. Increasing filtrations are standard in homology, decreasing in cohomology. Sources differ, and the indexing of the pages changes accordingly, so conventions should be checked before comparing formulas.
What happens if the filtration is not exhaustive?
The spectral sequence computes the homology of the union of the filtration stages, not of the whole complex. Exhaustiveness is what connects the answer to the intended target.
Can a spectral sequence collapse and still be uninformative?
Yes, if the extension problems remain unsolved. Collapse at E2 means all differentials vanish, which determines the associated graded exactly — but the target may still admit several non-isomorphic filtrations with that graded object.
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