CorollaryStatement: AI-adaptedProof: AI-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.

A split short exact sequence admits the direct-sum resolution

Statement

Assume the Axiom of Dependent Choice.

Let

0 \to A^{'} \to A \to A^{''} \to 0

be a split short exact sequence, and let projective resolutions of $A^{'}$ and $A^{''}$ be given. Then the sequence admits a projective resolution of $A$ whose degree- $n$ term is the direct sum $P_{n}^{'} \oplus P_{n}^{''}$ of the chosen side terms.

Facts & Assumptions

Given: A split short exact sequence $0 \to A^{'} \to A \to A^{''} \to 0$ and chosen projective resolutions of $A^{'}$ and $A^{''}$ .

[L1]

The horseshoe lemma produces a middle projective resolution (The horseshoe lemma for projective resolutions).

[L2]

A split short exact sequence identifies the middle object with the direct sum of the two ends (Split short exact sequence in an abelian category).

Proof

technique · direct

1.1

By [L2], identify $A$ with $A^{'} \oplus A^{''}$ . Applying [L1] to the chosen projective resolutions and the chosen splitting maps gives a middle resolution whose degree- $n$ term is $P_{n}^{'} \oplus P_{n}^{''}$ and whose augmentation is the direct-sum augmentation.

L1L2construct

2.1

Under that identification, the differentials are exactly the direct-sum differentials of the two side resolutions. Hence the split short exact sequence admits the direct-sum resolution.

step 1.1∎

Depends on

Used by

The horseshoe resolution of an extension of cyclic groups Example

Dependency tree · two levels

8 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, An Introduction to Homological Algebra (standard reference, not scraped)
Romyar Sharifi, Homological Algebra (standard reference, not scraped)