LemmaStatement: Literature-sourcedProof: AI-generatedprecheck passaudited 2026-09-01

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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.

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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

1.1L1L3L4construct

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 $0 \to \ker (ε^{'}) \to \ker (λ_{0}) \to \ker (ε^{''}) \to 0.$

2.1L2step 1.1∎

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

Depends on

Used by

The inductive horseshoe step Lemma

Dependency tree · two levels

27 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)