DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-30

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.

Chain complex in an abelian category

Definition

Let $A$ be an abelian category (Abelian category). A chain complex in $A$ is a family of objects $(C_{n})_{n \in Z}$ together with morphisms $d_{n} : C_{n} \to C_{n - 1}$ such that $d_{n - 1} \circ d_{n} = 0$ for every $n \in Z$ .

The pair is written $(C_{∙}, d)$ , or just $C_{∙}$ when the differentials are clear. The differential has degree $- 1$ .

Depends on

Abelian category

Used by

Bounded, bounded below, and bounded above complexes Definition
Chain map Definition
Cochain complex in an abelian category Definition
Cycle and boundary subobjects of a complex Definition
Exactness of a complex at a degree and acyclic complexes Definition
Subcomplex Definition
Zero complex and stalk complex Definition
FALSE: any sequence of morphisms is a chain complex False statement
The boundary subobject factors through the cycle subobject Lemma
The differential descends to a quotient complex Lemma
An additive functor applies degreewise to complexes and chain maps Proposition
An exact sequence is a complex, and its exactness agrees with the earlier notion Proposition
Products and coproducts of complexes are degreewise when they exist and preserve differentials Proposition

Dependency tree · two levels

3 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, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)
The Stacks Project, Section 12.13: Complexes (standard reference, not scraped)