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.

The category of chain complexes

Definition

The category of chain complexes in an abelian category $A$ is denoted $Ch (A) .$ Its objects are chain complexes in $A$ , and its morphisms are chain maps.

By Identities and composites of chain maps are chain maps, identities and composition are defined degreewise.

The full subcategories on bounded below, bounded above, and bounded complexes are denoted ${Ch}^{+} (A), {Ch}^{-} (A), {Ch}^{b} (A) .$

Depends on

Used by

Short exact sequence of complexes Definition
Isomorphisms of complexes are quasi-isomorphisms Proposition

Dependency tree · two levels

5 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)