LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05‡ 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 construction stays short exact after applying a right exact functor

Statement

Assume the Axiom of Dependent Choice.

Let $F : A \to B$ be a right exact functor between abelian categories, and let $0 \to A^{'} \to A \to A^{''} \to 0$ be a short exact sequence in $A$ . If $0 \to P_{∙}^{'} \to H_{∙} \to P_{∙}^{''} \to 0$ is a horseshoe short exact sequence of projective resolutions of $A^{'}, A, A^{''}$ , then $0 \to F (P_{∙, d e l}^{'}) \to F (H_{∙, d e l}) \to F (P_{∙, d e l}^{''}) \to 0$ is a short exact sequence of complexes in $B$ .

Facts & Assumptions

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

[L1]

A right exact functor is additive (A left or right exact functor between abelian categories is automatically additive).

[L2]

The horseshoe lemma produces a degreewise split short exact sequence of projective resolutions (The horseshoe lemma for projective resolutions).

[L3]

Additive functors apply degreewise to complexes and chain maps (An additive functor applies degreewise to complexes and chain maps).

[L4]

Exactness of a sequence of complexes is equivalent to exactness in each degree, and that is the definition of a short exact sequence of complexes (A sequence of chain maps is exact exactly when it is exact degreewise, Short exact sequence of complexes).

Proof

technique · direct

1.1

By [L2], each degree of the horseshoe row is a split short exact sequence $0 \to P_{n}^{'} \to H_{n} \to P_{n}^{''} \to 0.$ Because $F$ is additive by [L1], it preserves the biproduct decomposition carried by that split sequence, so each degree remains exact after applying $F$ .

L1L2givenalgebra

2.1

By [L3], the degreewise images from step 1.1 assemble into a sequence of chain maps $0 \to F (P_{∙, d e l}^{'}) \to F (H_{∙, d e l}) \to F (P_{∙, d e l}^{''}) \to 0.$ Since it is exact in every degree, [L4] identifies it as a short exact sequence of complexes.

L3L4step 1.1construct∎

Depends on

Used by

The connecting map for left derived functors Definition

Dependency tree · two levels

23 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, Chapter 2 `Derived Functors` (standard reference, not scraped)
Romyar Sharifi, Homological Algebra (standard reference, not scraped)