LemmaStatement: 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.

Exactness at the target of the connecting map

Statement

For a short exact sequence of complexes $0 \to A_{∙} \to B_{∙} \to C_{∙} \to 0,$ one has $im (\partial_{n} : H_{n} (C) \to H_{n - 1} (A)) = \ker (H_{n - 1} (A) \to H_{n - 1} (B)) .$

Facts & Assumptions

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

[L1]

Applying the weaker snake lemma to the quotient-kernel diagram in degree $n$ gives an exact segment $H_{n} (C) \overset{\partial_{n}}{\to} H_{n - 1} (A) \to H_{n - 1} (B)$ under the canonical kernel and cokernel identifications (The cycle-boundary diagram associated to a short exact sequence of complexes, Snake lemma under the weaker Stacks hypotheses, The connecting morphism in homology).

Proof

technique · direct

1.1

The three maps in [L1] are exactly the maps appearing in the statement.

L1given

2.1

Exactness of that categorical segment gives $im (\partial_{n}) = \ker (H_{n - 1} (A) \to H_{n - 1} (B)),$ as required.

L1step 1.1∎

Depends on

Used by

The long exact sequence in homology Theorem

Dependency tree · two levels

18 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)