False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-09-01

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FALSE: the degree-zero horseshoe lift is unique

Statement

False. Once the side resolutions and the short exact sequence are fixed, the lift $s : P_{0}^{''} \to A$ of the right augmentation through the middle epimorphism in the degree-zero horseshoe step is unique.

Facts & Assumptions

Given: The split short exact sequence $0 ⟶ Z \overset{x \mapsto (x, 0)}{\to} Z \oplus Z \overset{(x, y) \mapsto y}{\to} Z ⟶ 0,$ with each end object resolved by its length-zero identity resolution.

[L1]

The degree-zero horseshoe construction chooses a lift of the right augmentation through the middle epimorphism (The degree-zero horseshoe lift).

Refutation

technique · direct

1.1givenalgebra

Let $p (x, y) = y$ . The identity augmentation $Z \to Z$ lifts through $p$ both by $s_{1} (y) = (0, y)$ and by $s_{2} (y) = (y, y)$ , since $p s_{1} = p s_{2} = {id}_{Z}$ . The induced degree-zero middle augmentations are respectively $λ_{1} (x, y) = (x, y)$ and $λ_{2} (x, y) = (x + y, y)$ .

2.1L1step 1.1∎

By step 1.1, both maps satisfy the lifting equation required in [L1], but $s_{1} (1) = (0, 1) \neq (1, 1) = s_{2} (1)$ . Thus the degree-zero horseshoe lift is not unique.

Depends on

The degree-zero horseshoe lift

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

5 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)