Alexandrov–Čech cohomology

E173178

cohomology theory topological invariant

Alexandrov–Čech cohomology is a topological cohomology theory that computes invariants of spaces using inverse limits over open covers, closely related to and often coinciding with sheaf cohomology.

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All labels observed (3)

Label	Occurrences
Alexandrov–Čech cohomology canonical	1
sheaf cohomology	1
Čech cohomology	1

Statements (47)

Predicate	Object
instanceOf	cohomology theory ⓘ topological invariant ⓘ
alsoKnownAs	Alexandrov–Čech cohomology ⓘ surface form: Čech cohomology
appliesTo	locally contractible spaces ⓘ paracompact spaces ⓘ topological spaces ⓘ
closelyRelatedTo	Čech homology ⓘ
coefficientSystems	abelian groups ⓘ modules ⓘ sheaves of abelian groups ⓘ
coincidesWith	sheaf cohomology on paracompact spaces ⓘ sheaf cohomology with constant coefficients on good spaces ⓘ singular cohomology on reasonable spaces ⓘ
computes	cohomology groups of topological spaces ⓘ
constructionMethod	direct limit over open covers ⓘ inverse limit over refinements of open covers ⓘ
definedUsing	cochains on the nerve of an open cover ⓘ cocycles and coboundaries ⓘ
domain	topology ⓘ
field	algebraic topology ⓘ
generalizes	nerve-based computations of cohomology ⓘ
hasFeature	Mayer–Vietoris sequence in de Rham cohomology ⓘ surface form: Mayer–Vietoris sequence excision property under suitable conditions ⓘ homotopy invariance under suitable hypotheses ⓘ long exact sequence of a pair ⓘ
historicalPeriod	20th century mathematics ⓘ
namedAfter	Eduard Čech ⓘ Pavel Alexandrov ⓘ
partOf	homological algebra in topology ⓘ
relatedTo	Alexander–Spanier cohomology ⓘ sheaf cohomology ⓘ simplicial cohomology ⓘ singular cohomology ⓘ
satisfies	Eilenberg–Steenrod axioms up to mild conditions ⓘ
usedFor	computing cohomology of CW complexes ⓘ computing cohomology of manifolds ⓘ computing sheaf cohomology ⓘ defining cohomological invariants in geometry ⓘ studying local-to-global properties of spaces ⓘ
uses	cochain complexes ⓘ direct limits ⓘ inverse limits ⓘ nerve of an open cover ⓘ open covers ⓘ
usesConcept	direct system of cochain complexes ⓘ inverse system of cohomology groups ⓘ refinement of covers ⓘ

Referenced by (3)

Full triples — surface form annotated when it differs from this entity's canonical label.

Pavel Alexandrov → notableFor → Alexandrov–Čech cohomology ⓘ

Alexandrov–Čech cohomology → alsoKnownAs → Alexandrov–Čech cohomology ⓘ

this entity surface form: Čech cohomology

étale cohomology → definedUsing → Alexandrov–Čech cohomology ⓘ

this entity surface form: sheaf cohomology