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

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The Hom complex of chain complexes

Definition

Let $C_{∙}$ and $D_{∙}$ be chain complexes in an abelian category $A$ . Their Hom complex $\underset{}{Hom} (C, D)_{∙}$ is the chain complex of abelian groups whose degree- $r$ term is $\underset{}{Hom} (C, D)_{r} : = \prod_{n \in Z} {Hom}_{A} (C_{n}, D_{n + r}),$ so a degree- $r$ element is exactly a graded morphism $u = (u_{n})_{n}$ of degree $r$ .

The differential $\partial_{r} : \underset{}{Hom} (C, D)_{r} \to \underset{}{Hom} (C, D)_{r - 1}$ is defined componentwise by $(\partial u)_{n} = d_{n + r}^{D} u_{n} - (- 1)^{r} u_{n - 1} d_{n}^{C} .$

By The Hom-complex differential squares to zero ↗, this formula makes $\underset{}{Hom} (C, D)_{∙}$ into a chain complex.

Depends on

Used by

A chain homotopy Definition
A null-homotopic chain map Definition
The Hom complex of two two-term complexes Example
The Hom-complex differential squares to zero Lemma
Zero cocycles in the Hom complex are chain maps Proposition
Hom in the homotopy category is zero-degree homology of the Hom complex Theorem

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

Charles A. Weibel, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)
The Stacks Project, Section 12.16: Graded objects (standard reference, not scraped)