The Ladder of an Exact Couple and Rees Systems

Executive summary

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 F_p−1C ⊆ F_pC 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

D = ⊕_p H(F_pC), E = ⊕_p H(gr_p C)

The maps i, j, k are induced by the inclusion, the projection to the graded piece, and the connecting homomorphism respectively.

The couple is a compression of the ladder

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.

What the system tracks
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/F_pC)
lim¹ of the system	The obstruction to the comparison being an isomorphism

Why the formalism is worth having

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

0 → lim¹ H_n+1(C/F_p) → H_n(lim C/F_p) → lim H_n(C/F_p) → 0

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.

This is the standard trap in unbounded cases

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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