DefinitionDefinition: Literature-sourcedProof: Not applicablejudge 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.

Augmentation-preserving maps of projective resolutions

Definition

Let $\dots \to P_{1} \to P_{0} \overset{ε_{P}}{\to} A \to 0$ and $\dots \to Q_{1} \to Q_{0} \overset{ε_{Q}}{\to} B \to 0$ be projective resolutions, and let $u : A \to B$ be a morphism.

An augmentation-preserving map of projective resolutions lifting $u$ is a chain map $f_{∙} : P_{∙} \to Q_{∙}$ such that $ε_{Q} f_{0} = u ε_{P} .$

Thus the degree-zero square with the augmentations commutes, and the higher maps are compatible with the differentials because $f_{∙}$ is a chain map.

Augmentation-preserving maps of projective resolutions

Definition

Depends on

Dependency tree · 2 levels

Sources