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

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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.

Depends on

Used by

A chain homotopy Definition
The Hom complex of chain complexes Definition

Dependency tree · two levels

6 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

Joseph J. Rotman, An Introduction to Homological Algebra, 2nd ed. (standard reference, not scraped)
The Stacks Project, Section 12.16: Graded objects (standard reference, not scraped)