Filtered Differential Objects

Executive summary

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 … ⊆ F_p−1C ⊆ F_pC ⊆ … with associated graded

gr_p C = F_pC / F_p−1C

Conditions on a filtration
Condition	Meaning	Why it matters
Bounded	F_pC = 0 for small p and = C for large p, in each degree	Guarantees convergence with no further hypotheses
Exhaustive	The union of all F_pC is C	Necessary for the sequence to see all of C
Hausdorff	The intersection of all F_pC is 0	Prevents information being invisible to the filtration
Complete	C is the inverse limit of C/F_pC	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 04E_rCorrects 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.

The one-sentence summary

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 F_pH_n / F_p−1H_n where n = p + q. Reconstructing H_n requires solving a chain of extension problems.

Two groups with the same graded pieces need not be isomorphic

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.

NavigateContinue in this stream

Curated next steps from this page. The site also surfaces algorithmically related reading below.

ProvenanceSources and further reading

This page is an original KEVOS explanatory article. It presents the underlying mathematics — definitions, algorithms, complexity results and selection criteria — in KEVOS editorial voice. No text is reproduced from any copyrighted source. Where numerical tables are relevant, KEVOS links to live authoritative databases rather than republishing static values.