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 map

Definition

Let $C_{∙}$ and $D_{∙}$ be chain complexes. A chain map $f : C_{∙} \to D_{∙}$ is a family of morphisms $f_{n} : C_{n} \to D_{n}$ such that for every $n \in Z$ , $d_{n}^{D} \circ f_{n} = f_{n - 1} \circ d_{n}^{C} .$

Equivalently, each square $\begin{array}{ccc} C_{n} & \overset{f_{n}}{\to} & D_{n} \\ d_{n}^{C} ↓ & ↓ d_{n}^{D} \\ C_{n - 1} & \overset{f_{n - 1}}{\to} & D_{n - 1} \end{array}$ commutes.

Depends on

Chain complex in an abelian category

Used by

The category of chain complexes Definition
A chain map carries cycles to cycles and boundaries to boundaries Lemma
The cokernel of a chain map is computed degreewise Lemma
The kernel of a chain map is computed degreewise Lemma
An additive functor applies degreewise to complexes and chain maps Proposition
Identities and composites of chain maps are chain maps Proposition
The category of complexes in an additive category is additive Theorem

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