ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-12

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A filtered complex collapsing at e one

Example

Let $C_{1} = C_{0} = Z$ , $d_{1} = 2$ , with all other terms zero and the trivial filtration $F_{p} C = 0$ for p<0, $F_{p} C = C$ for p≥0. This spectral sequence collapses at $E^{1}$ , with the single stable term $E_{0, 0}^{1} = Z / 2$ .

Facts & Assumptions

Given: The trivial filtration on the multiplication-by-two complex.

[F1]

Pages use the filtered numerator/denominator formula (R page of the spectral sequence of a filtered complex).

[F2]

The initial differential is the graded map induced by d (The filtered differential induces d r on the r page).

[F3]

Finite convergence gives the image-filtered homology (Bounded filtered complex spectral sequence abuts to filtered homology).

[F4]

The map 2:ℤ→ℤ has zero kernel and cokernel ℤ/2 (Abelian-group model for spectral-sequence computations).

Verification

technique · direct

1.1

The initial page has ℤ at (0,1) and (0,0). Both generators have filtration degree 0, so $d^{0}$ is exactly multiplication by 2 between them. Thus its kernel at (0,1) is zero and its cokernel at (0,0) is ℤ/2 by [F4]. The r=1 formula [F1] gives these same quotients: the degree-one numerator is ker(2)=0 and the degree-zero denominator is 2ℤ.

F1F2F4

2.1

For every r≥1, $F_{- r} C = 0$ , so the degree-one numerator stays zero; $F_{r - 1} C_{1} = Z$ makes the degree-zero denominator stay 2ℤ. All other pieces are zero. Thus all differentials from $E^{1}$ onward have zero source or target and the sequence collapses there. Its homology is $H_{0} = Z / 2$ with $F_{- 1} = 0, F_{0} = H_{0}$ and $H_{n} = 0$ for n≠0, agreeing with [F3].

F1F3F4step 1.1∎

Source notes

Weibel, Chapter 5, Construction 5.4.6 and Lemma 5.4.7, pp.133–134; Sharifi, Theorem 4.2.3, pp.91–92. Increasing homological indices are used here.

Depends on

Used by

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Dependency tree · two levels

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Sources

Charles A. Weibel, An Introduction to Homological Algebra, Chapter 5 (standard reference, not scraped)
Romyar Sharifi, Homological Algebra (standard reference, not scraped)