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 injective 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 injective resolutions of $A^{'}$ and $A^{''}$ be given. Then there exists an injective resolution of $A$ whose degree- $n$ term is a finite product, equivalently biproduct, $I^{' n} \oplus I^{'' n}$ .

Facts & Assumptions

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

[L1]

The projective horseshoe lemma holds (The horseshoe lemma for projective resolutions).

[L2]

Injective resolutions are the cochain objects to be dualized (Injective resolutions in an abelian category).

[L3]

The opposite of an abelian category is abelian (The opposite of an abelian category is abelian).

Proof

technique · direct

1.1

By [L3], pass to the opposite abelian category. The given injective resolutions from [L2] become projective resolutions there, so [L1] supplies the dual horseshoe resolution in the opposite category.

L1L2L3construct

2.1

Translating back to the original category reverses arrows again and turns the opposite-category coproducts into finite products, which in an abelian category are the same biproducts. Hence one obtains the asserted injective horseshoe resolution.

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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)