PropositionStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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A short exact sequence of complexes gives six-term exact sequences when homology is concentrated in two degrees

Statement

Let $0 \to A_{∙} \to B_{∙} \to C_{∙} \to 0$ be a short exact sequence of complexes in an abelian category. Fix $n \in Z$ , and assume that all three complexes have zero homology outside degrees $n$ and $n - 1$ . Then the long exact sequence collapses to a six-term exact sequence $0 \to H_{n} (A) \to H_{n} (B) \to H_{n} (C) \overset{\partial_{n}}{\to} H_{n - 1} (A) \to H_{n - 1} (B) \to H_{n - 1} (C) \to 0.$

Facts & Assumptions

Given: A short exact sequence of complexes in an abelian category whose homology is concentrated in degrees $n$ and $n - 1$ .

[L1]

Every short exact sequence of complexes yields a long exact homology sequence (The long exact sequence in homology).

Proof

technique · direct

1.1L1givenconstruct

By [L1], the short exact sequence gives a long exact sequence running through the six displayed terms.

2.1step 1.1givenalgebra∎

Every homology term immediately before and after those six terms is zero by the concentration hypothesis. Removing those zero terms leaves the displayed six-term exact sequence.

Depends on

The long exact sequence in homology

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)