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

1.1L2givenconstruct

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 $ε_{Q} f_{0} = u ε_{P}$ .

2.1L1step 1.1∎

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

Lifting a map through degree zero of a projective resolution

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources