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.

Cochain complex in an abelian category

Definition

Let $A$ be an abelian category. A cochain 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 $n \in Z$ .

This page uses the reindexing convention that a cochain complex $C^{∙}$ may also be read as the chain complex with $(C^{♯})_{n} : = C^{- n}$ and $d_{n}^{♯} : = d^{- n}$ , so that lower and upper indices differ only by the sign of the grading.

Depends on

Used by

Bounded, bounded below, and bounded above complexes Definition
Cochain map Definition
Cohomology object of a cochain complex Definition
A chain map is a quasi-isomorphism exactly when its cochain reindexing is Proposition

Dependency tree · two levels

4 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)
Romyar Sharifi, Homological Algebra, §2.7 (standard reference, not scraped)