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

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Cohomology object of a cochain complex

Definition

Let $C^{∙}$ be a cochain complex. Its $n$ th cocycle subobject is $Z^{n} (C) : = \ker (d^{n}) ↪ C^{n},$ and its $n$ th coboundary subobject is $B^{n} (C) : = im (d^{n - 1}) ↪ C^{n} .$

Reading $C^{∙}$ as a chain complex by the reindexing convention of Cochain complex in an abelian category, the morphism $d^{n} d^{n - 1} = 0$ is exactly the chain-complex relation in degree $- n$ . Hence The boundary subobject factors through the cycle subobject supplies a canonical monomorphism $B^{n} (C) \to Z^{n} (C)$ .

The $n$ th cohomology object is its cokernel: $H^{n} (C) : = coker (B^{n} (C) \to Z^{n} (C)) .$

Equivalently, $H^{n} (C) = Z^{n} (C) / B^{n} (C)$ .

Depends on

Used by

Euler characteristic in a proper flat family is locally constant Corollary
Upper semicontinuity of fibre cohomology dimensions Corollary
Equal Euler classes do not by themselves prove homotopy-equivalent complexes Counterexample
Canonical truncation of a complex Definition
De rham cohomology Definition
First group cohomology via inhomogeneous one-cocycles Definition
Fixed-cover Čech cohomology Definition
Hochschild hyperhomology of a bounded bimodule complex Definition
K-flat complexes of abelian sheaves in the bounded-above setting Definition
Lie algebra cohomology Definition
Real singular cohomology Definition
Right derived objects relative to supplied injective resolution data Definition
Sheaf cohomology as right derived global sections Definition
Sheaf Ext of coherent modules Definition
Termwise Hochschild homology and iterated homology Definition
Cohomology of the empty space and the empty cover Example
Derived tensor product of abelian sheaves Lemma
Finite projective complex for proper flat coherent cohomology Lemma
Finite-free local criterion for cohomology and base change Lemma
Flat field extension commutes with coherent cohomology Lemma
Flat resolutions of abelian sheaves and K-flatness of bounded-above flat complexes Lemma
Local-section formula for derived direct image Lemma
Pushforward along a closed immersion preserves sheaf cohomology Lemma
Sheaf cohomology classes as derived morphisms Lemma
Universal finite projective cohomology complex over any base Lemma
A chain map is a quasi-isomorphism exactly when its cochain reindexing is Proposition
Cohomology and base change for proper flat coherent families Theorem
G0 of an abelian category equals triangle K0 of its bounded derived category Theorem
Refinement choices induce the same Čech map Theorem
Termwise Hochschild cyclicity for bounded projective bimodule complexes Theorem
Termwise Hochschild homology respects bimodule chain homotopies Theorem
The long exact sequence in cohomology Theorem
The zero-th right derived functor of a left exact functor recovers the functor Theorem

Dependency tree · two levels

9 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, Definition 2.7.7 (standard reference, not scraped)