A computation delivered one page at a time
A spectral sequence is a sequence of pages, each the homology of the previous one with respect to a differential, converging to a graded object of interest. It arises whenever a computation cannot be done in one step: a filtration, a double complex, or a composite of functors. Exact couples are the cleanest generating mechanism — an exact triangle of graded objects, repeatedly derived — and they make the existence of the whole apparatus a short construction.
Learning objectives
- Define a spectral sequence, its pages and differentials.
- Construct an exact couple and its derived couple.
- Identify the standard sources of spectral sequences.
- Explain what convergence does and does not deliver.
Section 01Pages and differentials
A spectral sequence is a family Erp,q with differentials
and Er+1 is the homology of Er under dr. The differentials grow longer with each page, so in a first-quadrant sequence each entry is eventually untouched: the sequence degenerates and E∞ is reached.
E∞ is the associated graded of a filtration of the target, NOT the target itself. Recovering the target requires solving extension problems — one for each filtration step. A spectral sequence that collapses completely still leaves this final step.
Section 02Exact couples
An exact couple is a pair of graded objects with three maps forming an exact triangle:
- Set d = j ∘ k: E → E. Exactness gives dd = jkjk = 0, since kj = 0.
- Set E′ = ker d / im d, the homology of E.
- Set D′ = i(D), the image of the first map.
- Define i′ as the restriction of i, j′ induced by j ∘ i−1, and k′ induced by k.
- The result (D′, E′) is again an exact couple. Iterating produces the pages of a spectral sequence.
They separate the formal machinery from the specific application. Once the derivation is understood, constructing the Grothendieck or Lyndon–Hochschild–Serre spectral sequence reduces to exhibiting the right couple.
Section 03Where they come from
| Source | Spectral sequence |
|---|---|
| A filtered chain complex | Converges to the homology of the complex |
| A double complex, two filtrations | Two spectral sequences with the same target |
| A composite of functors G ∘ F | The Grothendieck spectral sequence |
| A normal subgroup | Lyndon–Hochschild–Serre |
| A fibration of spaces | The Serre spectral sequence |
| A covering or open cover | Čech-to-derived-functor |
| A filtered ring or module | The associated graded spectral sequence |
The E2 page is normally computable; the differentials after it are not. Most practical work consists of arguing that particular differentials vanish — for dimension reasons, by naturality, or by comparison with a known case — rather than computing them. A spectral sequence is a constraint, not an algorithm.
ReferenceFrequently asked questions
Is a spectral sequence an algorithm?
No. It organises a computation and constrains the answer, but the differentials beyond the second page are generally unknown and must be determined by other means. Calling it a machine overstates what it delivers.
What does E<sub>∞</sub> mean when the sequence does not degenerate?
For each bidegree, the value the entry eventually stabilises to. In a first-quadrant sequence every entry stabilises after finitely many pages; in unbounded cases stabilisation requires a convergence hypothesis.
Why do the differentials change direction with the page?
Because each derivation shifts the filtration degree by one more step. The pattern is forced by the construction of the derived couple, and the bidegree of dr is the same for every spectral sequence of a given type.
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