DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-08-31

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.

A graded morphism of chain complexes

Definition

Let $C_{∙}$ and $D_{∙}$ be chain complexes in an abelian category $A$ , and let $r \in Z$ . A graded morphism of degree $r$ $u : C_{∙} \to D_{∙}$ is a family of morphisms $u_{n} : C_{n} \to D_{n + r} (n \in Z) .$

Thus degree $0$ graded morphisms are exactly degreewise families $u_{n} : C_{n} \to D_{n}$ , while degree $1$ graded morphisms shift the target one place to the left in homological degree.

A graded morphism of chain complexes

Definition

Depends on

Dependency tree · 2 levels

Sources