PropositionStatement: Literature-sourcedProof: AI-generatedPipeline-generatedprecheck passaudited 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 connecting morphism vanishes for a chain-split short exact sequence

Statement

Let $0 \to A_{∙} \overset{i}{\to} B_{∙} \overset{p}{\to} C_{∙} \to 0$ be a short exact sequence of complexes. If it admits either a chain section $s : C_{∙} \to B_{∙}$ with $p s = 1_{C}$ or a chain retraction $r : B_{∙} \to A_{∙}$ with $r i = 1_{A}$ , then $\partial_{n} = 0 : H_{n} (C) \to H_{n - 1} (A)$ for every $n$ .

Facts & Assumptions

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

[L1]

A chain map is a degreewise morphism commuting with the differentials (Chain map).

[L2]

The connecting morphism is induced from the preconnecting arrow on cycles (The connecting morphism in homology, The preconnecting arrow on cycles).

[L3]

The associated homology sequence is exact (The long exact sequence in homology).

Proof

technique · direct

1.1

Suppose first that there is a chain section $s$ with $p s = 1_{C}$ . Functoriality of homology gives $H_{n} (p) H_{n} (s) = 1_{H_{n} (C)}$ , so $H_{n} (p)$ is epic. Exactness in [L3] gives $\ker (\partial_{n}) = im (H_{n} (p)) = H_{n} (C)$ , hence $\partial_{n} = 0$ .

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2.1

Suppose instead that there is a chain retraction $r$ with $r i = 1_{A}$ . Then $H_{n - 1} (r) H_{n - 1} (i) = 1_{H_{n - 1} (A)}$ , so $H_{n - 1} (i)$ is monic. Exactness in [L3] gives $im (\partial_{n}) = \ker (H_{n - 1} (i)) = 0$ , and a morphism with zero image in an abelian category is zero. Thus the connecting morphism vanishes in either chain-split situation.

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Depends on

Used by

Homology of a chain-split direct-sum sequence Corollary

Dependency tree · two levels

14 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)