CorollaryStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passaudited 2026-09-01

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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A short exact sequence with acyclic middle complex identifies neighbouring homology

Statement

If $0 \to A_{∙} \to B_{∙} \to C_{∙} \to 0$ is a short exact sequence of complexes and $B_{∙}$ is acyclic, then each connecting morphism $\partial_{n} : H_{n} (C) \tilde{\to} H_{n - 1} (A)$ is an isomorphism.

Facts & Assumptions

Given: A short exact sequence $0 \to A_{∙} \to B_{∙} \to C_{∙} \to 0$ of complexes with $B_{∙}$ acyclic.

[L1]

The long exact homology sequence exists (The long exact sequence in homology).

[L2]

Acyclic means that every homology object of the complex is zero (Exactness of a complex at a degree and acyclic complexes).

Proof

technique · direct

1.1L1givenconstruct

By [L1], the degree- $n$ window is $H_{n} (B) \to H_{n} (C) \overset{\partial_{n}}{\to} H_{n - 1} (A) \to H_{n - 1} (B) .$

2.1L1L2step 1.1algebra∎

The outer terms in that window are zero by [L2]. Exactness then forces $\partial_{n}$ to be both monic and epic, hence an isomorphism.

Depends on

Used by

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Dependency tree · two levels

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Sources

Charles A. Weibel, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)
The Stacks Project, Section 12.13: Complexes (standard reference, not scraped)