LemmaStatement: Literature-sourcedProof: AI-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.

The inductive horseshoe step

Statement

Suppose $0 \to K_{n}^{'} \to K_{n} \to K_{n}^{''} \to 0$ is the short exact sequence of current kernels arising in the horseshoe construction. If the tails of projective resolutions of $K_{n}^{'}$ and $K_{n}^{''}$ are already chosen, then one more degree of the horseshoe construction produces a projective object $P_{n + 1} = P_{n + 1}^{'} \oplus P_{n + 1}^{''}$ surjecting onto $K_{n}$ and a new short exact sequence of next kernels.

Facts & Assumptions

Given: The current short exact kernel sequence and the next projective terms of the two side resolutions.

[L1]

The degree-zero horseshoe construction produces the next surjection from a direct sum of projectives (The degree-zero horseshoe lift).

[L2]

The kernel of that new surjection again sits in a short exact sequence with the two side kernels (The horseshoe kernel fits into a short exact sequence).

Proof

technique · direct

1.1

Apply [L1] to the short exact sequence $0 \to K_{n}^{'} \to K_{n} \to K_{n}^{''} \to 0$ and to the degree-zero tails of the already chosen projective resolutions of $K_{n}^{'}$ and $K_{n}^{''}$ . This produces a projective object $P_{n + 1} = P_{n + 1}^{'} \oplus P_{n + 1}^{''}$ together with an epimorphism $P_{n + 1} ↠ K_{n}$ .

L1givenconstruct

2.1

Applying [L2] to that new surjection gives the next short exact sequence of kernels. Therefore the horseshoe construction advances by one degree.

L2step 1.1∎

Depends on

Used by

The horseshoe lemma for projective resolutions Theorem

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)