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

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.

The degree-zero horseshoe lift

Statement

Let $0 \to A^{'} \overset{i}{\to} A \overset{p}{\to} A^{''} \to 0$ be a short exact sequence, and let $P_{0}^{'} ↠ A^{'}, P_{0}^{''} ↠ A^{''}$ be the degree-zero terms of projective resolutions of $A^{'}$ and $A^{''}$ . Then there exists an epimorphism $λ_{0} : P_{0}^{'} \oplus P_{0}^{''} ↠ A$ whose restrictions to the two summands are $i ε^{'}$ and a lift of $ε^{''}$ through $p$ . In particular $P_{0}^{'} \oplus P_{0}^{''}$ is projective.

Facts & Assumptions

Given: The short exact sequence above and projective resolutions of $A^{'}$ and $A^{''}$ .

[L1]

Projective resolutions provide the augmentations and projective degree-zero terms (Projective resolutions in an abelian category).

[L2]

Projective objects lift across epimorphisms (Projective object).

[L3]

Finite coproducts of projectives are projective (A coproduct of projectives is projective and a product of injectives is injective).

[L4]

In the displayed short exact sequence, $p$ is the cokernel of $i$ (Exact sequence and short exact sequence in an abelian category).

Proof

technique · direct

1.1L1L2construct

Since $p : A ↠ A^{''}$ is epic and $P_{0}^{''}$ is projective, [L2] lifts the augmentation $ε^{''} : P_{0}^{''} \to A^{''}$ to a map $s : P_{0}^{''} \to A$ with $p s = ε^{''}$ .

2.1L1L3L4step 1.1∎

Define $λ_{0} = [i ε^{'}, s] : P_{0}^{'} \oplus P_{0}^{''} \to A$ . To prove it epic without elements, let $c : A \to C$ satisfy $c λ_{0} = 0$ . Restricting to the first summand gives $c i ε^{'} = 0$ , so $c i = 0$ because $ε^{'}$ is epic. By [L4], $c$ factors as $c = c^{'} p$ for some $c^{'} : A^{''} \to C$ . Restricting to the second summand now gives $0 = c s = c^{'} p s = c^{'} ε^{''}$ , so $c^{'} = 0$ because $ε^{''}$ is epic. Thus $c = 0$ . In an abelian category the cokernel of $λ_{0}$ therefore vanishes, so $λ_{0}$ is epic. Its source is projective by [L3].

Depends on

Used by

FALSE: the degree-zero horseshoe lift is unique False statement
The horseshoe kernel fits into a short exact sequence Lemma
The inductive horseshoe step Lemma

Dependency tree · two levels

11 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)