TheoremStatement: 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.

The horseshoe lemma for projective resolutions

Statement

Assume the Axiom of Dependent Choice.

Let $0 \to A^{'} \to A \to A^{''} \to 0$ be a short exact sequence, and let projective resolutions of $A^{'}$ and $A^{''}$ be given. Then there exists a projective resolution of $A$ whose degree- $n$ term is $P_{n}^{'} \oplus P_{n}^{''}$ , fitting into a degreewise split short exact sequence of augmented complexes.

Facts & Assumptions

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

[L1]

The inductive horseshoe step advances the construction by one degree (The inductive horseshoe step).

[L2]

A projective resolution is an exact augmented complex of projective objects (Projective resolutions in an abelian category).

[L3]

Finite coproducts of projectives are projective (A coproduct of projectives is projective and a product of injectives is injective).

[L4]

Dependent choice licenses the countable successor-by-successor selection of the horseshoe lifts and kernel sequences (The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain).

Proof

technique · direct

1.1

Start at degree zero with the original short exact sequence $0 \to A^{'} \to A \to A^{''} \to 0.$ Every current short exact kernel sequence extends one more degree by [L1], and the next choice depends on the previously constructed degree. Therefore [L4] produces the whole augmented complex for $A$ , whose degree- $n$ term is $P_{n}^{'} \oplus P_{n}^{''}$ and whose kernel sequences remain short exact in every degree.

L1L4construct

2.1

Each term $P_{n}^{'} \oplus P_{n}^{''}$ is projective by [L3], and the short exact kernel sequences from step 1.1 are exactly the data needed for exactness of the middle augmented complex. Therefore [L2] identifies the resulting complex as a projective resolution of $A$ .

L2L3step 1.1∎

Depends on

Used by

Dependency tree · two levels

17 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)