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

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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

1.1L1L2L3construct

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.

2.1step 1.1∎

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

Projective comparison maps are unique up to chain homotopy

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources