DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-30

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Cochain complex in an abelian category

Definition

Let $A$ be an abelian category. A cochain complex in $A$ is a family of objects $(C^{n})_{n \in Z}$ together with morphisms $d^{n} : C^{n} \to C^{n + 1}$ such that $d^{n + 1} \circ d^{n} = 0$ for every $n \in Z$ .

This page uses the reindexing convention that a cochain complex $C^{∙}$ may also be read as the chain complex with $(C^{♯})_{n} : = C^{- n}$ and $d_{n}^{♯} : = d^{- n}$ , so that lower and upper indices differ only by the sign of the grading.

Depends on

Used by

Bounded graded bimodule complexes and signed tensor totalization Definition
Bounded, bounded below, and bounded above complexes Definition
Brutal truncation of a complex Definition
Cartan-Eilenberg injective resolution of a bounded-below complex Definition
Coaugmented cochain complexes under an object Definition
Cochain map Definition
Cohomological mixed Hodge complexes Definition
Cohomology object of a cochain complex Definition
De rham cochain complex Definition
Derived category of an abelian category Definition
K-flat complexes of abelian sheaves in the bounded-above setting Definition
Lie algebra cohomology Definition
Right derived objects relative to supplied injective resolution data Definition
Signed totalization of graded Aₘ-bimodule actions Definition
Tensor product of abelian sheaves and its total complex Definition
The Hom cochain complex of a chain complex Definition
Associator, symmetry and unitors of the abelian sheaf tensor product Lemma
Derived tensor product of abelian sheaves Lemma
Finite poset diagrams have bounded finite bar models Lemma
Flat resolutions of abelian sheaves and K-flatness of bounded-above flat complexes Lemma
K-flat sheaf complexes preserve quasi-isomorphisms Lemma
Lifting a morphism from an exact complex into an injective resolution Lemma
Locally split Cech nerves give actual derived Cartesian sheaf descent Lemma
Mixed Hodge complex cohomology and simplicial weights Lemma
Morphisms from the constant sheaf are global sections Lemma
Pushforward along a closed immersion preserves sheaf cohomology Lemma
Sheaf cohomology classes as derived morphisms Lemma
The induced cohomology map is independent of the chosen injective comparison extension Lemma
A chain map is a quasi-isomorphism exactly when its cochain reindexing is Proposition
A natural transformation induces natural transformations of right derived functors Proposition
Change-of-injective-resolution isomorphisms satisfy identity and cocycle laws Proposition
Long exact sequence of sheaf cohomology Theorem
Naturality of the cohomology connecting morphism Theorem
Refinement choices induce the same Čech map Theorem
Right derived functors relative to supplied data are additive functors Theorem
Termwise Hochschild spectral sequence of a bounded bimodule complex Theorem
The cohomological filtered complex construction Theorem
The long exact sequence in cohomology Theorem
Two supplied injective resolution data define naturally isomorphic right derived functors Theorem

Dependency tree · two levels

4 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)
Romyar Sharifi, Homological Algebra, §2.7 (standard reference, not scraped)