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.
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Projective comparison maps exist

Statement

Assume the Axiom of Dependent Choice.

Let $u : A \to B$ be a morphism, and let $P_{∙} \to A$ and $Q_{∙} \to B$ be projective resolutions. Then there exists an augmentation-preserving chain map $f_{∙} : P_{∙} \to Q_{∙}$ lifting $u$ .

Facts & Assumptions

Given: A morphism $u : A \to B$ and projective resolutions $P_{∙} \to A$ , $Q_{∙} \to B$ .

[L1]

Degree zero can be lifted across the target augmentation (Lifting a map through degree zero of a projective resolution).

[L2]

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

[L3]

The required notion is an augmentation-preserving chain map (Augmentation-preserving maps of projective resolutions).

[L4]

Dependent choice licenses the countable successor-by-successor selection of compatible lifts (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

By [L1], choose $f_{0} : P_{0} \to Q_{0}$ with $ε_{Q} f_{0} = u ε_{P}$ .

L1construct

2.1

Starting from the degree-zero lift in step 1.1, every partial comparison map through degree $n$ extends to one through degree $n + 1$ by [L2]. The successive choices depend on the previously chosen partial map, so [L4] produces maps $f_{n}$ in every degree.

L2L4step 1.1choose

3.1

The family $(f_{n})$ is a chain map by construction, and step 1.1 gives the augmentation identity. Hence [L3] is satisfied, so $f_{∙}$ is a comparison map lifting $u$ .

L3step 1.1step 2.1∎

Depends on

Used by

Comparison maps between two resolutions of a cyclic group Example
Comparison maps respect composition up to homotopy Proposition
Comparison of the identity is homotopic to the identity Proposition
Injective comparison maps exist Theorem
Projective comparison maps are unique up to chain homotopy Theorem
Projective resolutions of the same object are homotopy equivalent over that object Theorem

Dependency tree · two levels

15 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)