TheoremStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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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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Projective comparison maps are unique up to chain homotopy

Statement

Assume the Axiom of Dependent Choice.

Any two augmentation-preserving maps between projective resolutions lifting the same object morphism are chain-homotopic.

Facts & Assumptions

Given: Two augmentation-preserving maps $f_{∙}, g_{∙} : P_{∙} \to Q_{∙}$ between projective resolutions, lifting the same object morphism $u : A \to B$ .

[L1]

A partial comparison homotopy extends one degree at a time (Extending a partial comparison homotopy by one degree).

[L2]

The maps $f_{∙}$ and $g_{∙}$ are comparison maps in the sense of Augmentation-preserving maps of projective resolutions.

[L3]

Dependent choice licenses the countable successor-by-successor selection of compatible homotopy components (The axiom of dependent choice: a relation in which every element is related to something admits an $N$ -indexed chain).

Proof

technique · direct

1.1

Start at degree $0$ . Because $f_{∙}$ and $g_{∙}$ lift the same object map, [L1] produces $h_{0} : P_{0} \to Q_{1}$ . Every partial homotopy through degree $n - 1$ extends one degree further by [L1], and the successive choices depend on the previously chosen components. Therefore [L3] produces a family $h_{n} : P_{n} \to Q_{n + 1}$ in every degree.

L1L2L3construct

2.1

By construction, the family $(h_{n})$ satisfies the defining chain-homotopy equation in every degree. Therefore $f_{∙}$ and $g_{∙}$ are chain-homotopic.

step 1.1∎

Depends on

Used by

An explicit comparison homotopy Example
FALSE: a comparison map between resolutions is unique as a chain map False statement
Comparison maps respect composition up to homotopy Proposition
Horseshoe resolutions are compatible with morphisms of short exact sequences up to homotopy Proposition
Injective comparison maps are unique up to cochain homotopy Theorem
Projective resolutions of the same object are homotopy equivalent over that object Theorem

Dependency tree · two levels

18 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)