LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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Acyclic assembly by exact rows

Statement

Let $K^{p, q}$ be a first-quadrant double cochain complex whose signed total complex uses finite direct sums on every diagonal. Suppose a cochain complex $E^{∙}$ maps to the left edge so that, for every $q$ , the augmented row $0 ⟶ E^{q} ⟶ K^{0, q} \overset{h}{\to} K^{1, q} \overset{h}{\to} K^{2, q} ⟶ \dots$ is exact and the augmentations commute with the vertical maps. Then the induced cochain map $E^{∙} \to Tot K$ is a quasi-isomorphism.

Facts & Assumptions

Given: The first-quadrant double complex, compatible row augmentations, and exact augmented rows stated above.

Proof

technique · direct

1.1

Interchange the two indices of $K$ . Multiplying the component in bidegree $(p, q)$ by $(- 1)^{p q}$ identifies its signed total complex with the total complex after the interchange; the augmented rows become exact augmented columns with compatible edge maps.

givenalgebra

2.1

Apply the exact-column assembly lemma to the interchanged double complex. Transporting its quasi-isomorphism back through the sign identification gives $E^{∙} \to Tot K$ .

step 1.1algebra∎

Depends on

Used by

FALSE: balance of Ext requires spectral-sequence pages False statement
Projective and injective constructions of Ext agree for supplied resolutions Theorem
The bar cochain complex computes group cohomology Theorem

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Charles A. Weibel, An Introduction to Homological Algebra, Chapter 2 (standard reference, not scraped)