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

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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.

Projective resolutions of the same object are homotopy equivalent over that object

Statement

Assume the Axiom of Dependent Choice.

Any two projective resolutions of the same object are homotopy equivalent over that object.

Facts & Assumptions

Given: Two projective resolutions $P_{∙} \to A$ and $Q_{∙} \to A$ of the same object $A$ .

[L1]

Comparison maps between projective resolutions exist (Projective comparison maps exist).

[L2]

Two comparison maps lifting the same object morphism are chain-homotopic (Projective comparison maps are unique up to chain homotopy).

Proof

technique · direct

1.1

Apply [L1] to $1_{A}$ in each direction. This gives comparison maps $f_{∙} : P_{∙} \to Q_{∙}$ and $g_{∙} : Q_{∙} \to P_{∙}$ , both lifting the identity on $A$ .

L1construct

2.1

The composites $g_{∙} f_{∙}$ and $f_{∙} g_{∙}$ also lift $1_{A}$ , as do the identity chain maps on $P_{∙}$ and $Q_{∙}$ . By [L2], $g_{∙} f_{∙} ≃ 1_{P_{∙}} and f_{∙} g_{∙} ≃ 1_{Q_{∙}} .$ Thus the two resolutions are homotopy equivalent over $A$ , including when $A = 0$ .

L2step 1.1∎

Depends on

Used by

A projective or injective resolution is unique up to nonunique homotopy equivalence Corollary

Dependency tree · two levels

9 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)