Convergence is a hypothesis, not a conclusion
A spectral sequence converges when every entry stabilises and the stable page is the associated graded of a filtration of the intended target. For first-quadrant sequences with a bounded filtration this is automatic. Outside that setting it must be established, and the standard failure is a non-vanishing lim1 term: each page behaves correctly and the limit still fails to compute the target. Convergence statements always carry hypotheses, and they are not decorative.
Learning objectives
- State when convergence is automatic.
- Identify the role of boundedness and exhaustiveness.
- Recognise collapse and degeneration.
- Name the standard failure modes.
Section 01The easy case
If the spectral sequence is first-quadrant — entries vanish outside p, q ≥ 0 — then for each bidegree the differentials in and out vanish for large r, so the entry stabilises after finitely many pages. With a bounded filtration on the target, convergence follows.
Differentials on page r have bidegree (r, 1−r), so for large r either the source or the target leaves the quadrant. Every entry is therefore eventually untouched. Almost every spectral sequence met in a first course is of this type.
| Setting | Convergence |
|---|---|
| First-quadrant, bounded filtration | Automatic |
| Bounded double complex | Automatic |
| Half-plane with exiting differentials | Automatic |
| Half-plane with entering differentials | Needs completeness and vanishing lim1 |
| Unbounded | Conditional convergence at best; extra hypotheses required |
Section 02Collapse and degeneration
Er = E∞. The associated graded is then known, but the extension problem remains.
If E2 is concentrated in a single row or column, all differentials vanish for degree reasons AND the filtration has one step, so the target is determined exactly.
Balance of Ext is proved by concentration: one filtration leaves a single column, the other a single row, and both compute the same total homology. The five-term exact sequence is the next-best case, where only two positions contribute in low total degree.
Section 03Failure modes
Non-vanishing lim1
For unbounded filtrations, the inverse limit of the filtration quotients may not compute the target. A lim1 term intervenes and must be shown to vanish.
Unsolved extensions
The associated graded is known but the target is not determined. Common and often unavoidable without additional structure.
Unknown differentials
The E2 page is computable but a differential cannot be determined. The answer is then bounded above and below, not pinned down.
Non-exhaustive filtration
The sequence converges to the homology of the union of stages, which may be a proper part of the target.
Wrong convergence direction
Cohomological and homological conventions differ; applying a homological convergence statement to a cohomological sequence gives a false conclusion.
Conditional convergence only
Boardman's conditional convergence gives a weaker statement that becomes genuine convergence only under an additional vanishing hypothesis.
ReferenceFrequently asked questions
Does convergence guarantee I can compute the answer?
No. It guarantees the associated graded of a filtration of the answer. The extension problems and any unknown differentials remain, and both are frequently the hard part of a computation.
What is conditional convergence?
Boardman's notion for unbounded spectral sequences: the sequence converges conditionally when the relevant inverse limit and its first derived functor vanish appropriately. It becomes strong convergence under an additional hypothesis on the E∞ page.
How do I show a differential vanishes?
Common arguments: the source or target is zero for degree reasons; naturality with respect to a map where the answer is known; multiplicativity if the sequence is a ring; or comparison with a case computed independently.
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