CorollaryStatement: AI-adaptedProof: 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.

Homology of a chain-split direct-sum sequence

Statement

If $0 \to A_{∙} \to B_{∙} \to C_{∙} \to 0$ is a chain-split short exact sequence of complexes, then for every $n$ there is an isomorphism $H_{n} (B) ≅ H_{n} (A) \oplus H_{n} (C) .$

Facts & Assumptions

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

[L1]

In a chain-split short exact sequence, every connecting morphism is zero (The connecting morphism vanishes for a chain-split short exact sequence).

[L2]

Every short exact sequence of complexes has a long exact homology sequence (The long exact sequence in homology).

Proof

technique · direct

1.1

By [L1] and [L2], the long exact homology sequence breaks in each degree into a short exact sequence $0 \to H_{n} (A) \to H_{n} (B) \to H_{n} (C) \to 0.$

L1L2givenalgebra

2.1

Because the short exact sequence is chain split, the middle complex is degreewise isomorphic to $A_{∙} \oplus C_{∙}$ with block-diagonal differential. Hence $Z_{n} (B) ≅ Z_{n} (A) \oplus Z_{n} (C), B_{n} (B) ≅ B_{n} (A) \oplus B_{n} (C),$ and quotienting cycles by boundaries gives $H_{n} (B) ≅ H_{n} (A) \oplus H_{n} (C) .$

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

10 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)