LemmaStatement: Literature-sourcedProof: 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 kernel fits into a short exact sequence

Statement

With the notation of the degree-zero horseshoe lift, let $K : = \ker (λ_{0}), Ω^{1} (A^{'}) : = \ker (ε^{'}), Ω^{1} (A^{''}) : = \ker (ε^{''}) .$ Then there is a short exact sequence $0 \to Ω^{1} (A^{'}) \to K \to Ω^{1} (A^{''}) \to 0.$

Facts & Assumptions

Given: The degree-zero horseshoe map $λ_{0} : P_{0}^{'} \oplus P_{0}^{''} ↠ A$ from The degree-zero horseshoe lift.

[L1]

The degree-zero horseshoe lift gives a commutative diagram with exact rows (The degree-zero horseshoe lift).

[L2]

First syzygies are the kernels of the augmentations (Syzygies and cosyzygies relative to a chosen resolution).

[L3]

The snake lemma extracts an exact kernel sequence from a commutative short-exact diagram (Snake lemma in an abelian category).

[L4]

The nine-lemma package supplies the exactness compatibilities used in the ambient $3 \times 3$ diagram (Nine lemma in an abelian category).

Proof

technique · direct

1.1

The map $λ_{0}$ from [L1] fits into a commutative diagram $0 \to P_{0}^{'} \to P_{0}^{'} \oplus P_{0}^{''} \to P_{0}^{''} \to 0$ over $0 \to A^{'} \to A \to A^{''} \to 0,$ where both rows are exact and the top row is split exact. Applying the snake lemma [L3] gives an exact sequence $0 \to \ker (ε^{'}) \to \ker (λ_{0}) \to \ker (ε^{''}) \to 0.$

L1L3L4construct

2.1

By [L2], these three kernels are exactly $Ω^{1} (A^{'})$ , $K$ , and $Ω^{1} (A^{''})$ . Hence the displayed kernel sequence is short exact.

L2step 1.1∎

Depends on

Used by

The inductive horseshoe step Lemma

Dependency tree · two levels

24 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)