TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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The long exact sequence in homology

Statement

Let $0 \to A_{∙} \to B_{∙} \to C_{∙} \to 0$ be a short exact sequence of complexes in an abelian category. Then there is an exact sequence $\dots \to H_{n} (A) \to H_{n} (B) \to H_{n} (C) \overset{\partial_{n}}{\to} H_{n - 1} (A) \to H_{n - 1} (B) \to H_{n - 1} (C) \to \dots .$

Facts & Assumptions

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

[L1]

Every chain map induces a well-defined map on homology (A chain map induces a well-defined map on homology).

[L2]

The four local exactness claims around $\partial_{n}$ are provided by the preceding exactness lemmas (Exactness at the homology of the left complex, Exactness at the homology of the middle complex, Exactness at the homology of the right complex, Exactness at the target of the connecting map).

Proof

technique · direct

1.1L1givenconstruct

The maps $H_{n} (A) \to H_{n} (B)$ and $H_{n} (B) \to H_{n} (C)$ are defined by [L1], and the connecting map $\partial_{n}$ is defined by The connecting morphism in homology. Thus the displayed sequence exists in every degree.

2.1L2step 1.1algebra

For each integer $n$ , [L2] gives exactness at $H_{n} (A)$ , $H_{n} (B)$ , $H_{n} (C)$ , and $H_{n - 1} (A)$ . Therefore every four-term window around $\partial_{n}$ is exact.

3.1step 2.1algebra∎

Since $n$ was arbitrary, these exact windows concatenate to the displayed bi-infinite exact sequence.

The long exact sequence in homology

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources