Making the convergence bookkeeping explicit
An exact couple arising from a filtered complex carries more structure than the couple alone records: the filtration stages assemble into a ladder of long exact sequences, and tracking that ladder is what allows convergence to be proved rather than assumed. Rees systems package the ladder into a single algebraic object, so that questions about limits and completions become questions about the system.
Learning objectives
- Describe the ladder attached to a filtered complex.
- State what a Rees system records.
- Explain how the limit of the system relates to the target.
- Identify where the lim1 obstruction enters.
Section 01The ladder
Each inclusion Fp−1C ⊆ FpC gives a short exact sequence of complexes and hence a long exact homology sequence. Stacking these produces a ladder, with the exact couple assembling the two graded objects
The maps i, j, k are induced by the inclusion, the projection to the graded piece, and the connecting homomorphism respectively.
All the information used by the spectral sequence sits in the couple, but questions about whether the answer is the homology of C — rather than of some limit — require the ladder, because they concern how the filtration stages assemble.
Section 02Rees systems
A Rees system records the ladder together with the maps relating a filtered complex to its completion. It carries two exact couples and comparison maps between them, so that both the spectral sequence and the convergence question are visible in a single object.
| Component | Records |
|---|---|
| The exact couple | The spectral sequence and its pages |
| The direct limit | Whether the filtration is exhaustive |
| The inverse limit | Whether the filtration is complete |
| The comparison maps | How H(C) relates to the limit of the H(C/FpC) |
| lim1 of the system | The obstruction to the comparison being an isomorphism |
For bounded filtrations none of this is needed — convergence is immediate. The machinery earns its place for unbounded filtrations, completions, and pro-objects, where informal arguments give wrong answers.
Section 03Limits and the obstruction
The homology of the completed complex sits in a short exact sequence
so the answer computed by the spectral sequence is the right-hand term, and the lim1 contribution is invisible to it. When lim1 vanishes the two agree; when it does not, the spectral sequence converges to something other than the intended target.
Every page of the spectral sequence can be correct and the conclusion still wrong, because the target was never what the sequence converges to. Checking a Mittag-Leffler condition — which forces lim1 to vanish — is the usual remedy.
ReferenceFrequently asked questions
Do I need Rees systems to use spectral sequences?
For first-quadrant sequences, no — convergence is automatic and the couple suffices. The formalism matters when filtrations are unbounded or when completions are involved, which is common in stable homotopy theory and in pro-algebraic settings.
What is the Mittag-Leffler condition?
That the images of the maps in an inverse system stabilise. It implies lim1 vanishes and is the standard checkable hypothesis in convergence statements.
Is the ladder the same as the exact couple?
The couple is obtained from the ladder by taking direct sums over the filtration index. The ladder retains the individual sequences, which is what convergence arguments need.
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