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.

Augmented chain complexes over an object

Definition

Let $A$ be an object of an abelian category. An augmented chain complex over $A$ is a chain complex in nonnegative degrees $\dots \overset{d_{3}}{\to} P_{2} \overset{d_{2}}{\to} P_{1} \overset{d_{1}}{\to} P_{0}$ together with a morphism $ε : P_{0} \to A$ such that $ε d_{1} = 0.$

Equivalently, one has an extended chain $\dots \to P_{2} \to P_{1} \to P_{0} \to A \to 0$ whose consecutive composites are zero.

Depends on

Chain complex in an abelian category

Used by

Projective resolutions in an abelian category Definition
One-step extension of a partial projective resolution Lemma

Dependency tree · two levels

2 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)