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.

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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.

The Hom complex of chain complexes

Definition

Depends on

Dependency tree · 2 levels

Sources