TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01‡ rests on unproved material (inherited)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

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

1.1

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.

L1givenconstruct

2.1

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.

L2step 1.1algebra

3.1

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

step 2.1algebra∎

Depends on

Used by

Dependency tree · two levels

13 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)
The Stacks Project, Section 12.13: Complexes (standard reference, not scraped)