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

How statement and proof provenance work

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.

Lifting a map through degree zero of a projective resolution

Statement

Let $u : A \to B$ be a morphism, let $P_{∙} \to A$ be a projective resolution, and let $Q_{∙} \to B$ be a projective resolution. Then there exists a morphism $f_{0} : P_{0} \to Q_{0}$ such that $ε_{Q} f_{0} = u ε_{P} .$

Facts & Assumptions

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

[L1]

An augmentation-preserving comparison map is required to satisfy $ε_{Q} f_{0} = u ε_{P}$ at degree zero (Augmentation-preserving maps of projective resolutions).

[L2]

Projective objects lift across epimorphisms (Projective object).

Proof

technique · direct

1.1

The augmentation $ε_{Q} : Q_{0} ↠ B$ is epic, and $P_{0}$ is projective. Apply [L2] to the composite $u ε_{P} : P_{0} \to B$ to obtain a lift $f_{0} : P_{0} \to Q_{0}$ with $ε_{Q} f_{0} = u ε_{P}$ .

L2givenconstruct

2.1

By [L1], the map $f_{0}$ of step 1.1 is exactly the required degree-zero part of an augmentation-preserving comparison map.

L1step 1.1∎

Depends on

Used by

Extending a partial comparison map by one degree Lemma
Projective comparison maps exist Theorem

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)