étale cohomology

E254118

cohomology theory mathematical concept tool in algebraic geometry

Étale cohomology is a cohomology theory in algebraic geometry that allows one to apply topological and cohomological methods to schemes, particularly over fields with nontrivial arithmetic such as finite fields.

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All labels observed (2)

Label	Occurrences
étale cohomology canonical	2
foundations of étale cohomology	1

Statements (67)

Predicate	Object
instanceOf	cohomology theory ⓘ mathematical concept ⓘ tool in algebraic geometry ⓘ
analogOf	singular cohomology for schemes ⓘ
appliesTo	algebraic varieties ⓘ schemes ⓘ schemes over fields with nontrivial arithmetic ⓘ schemes over finite fields ⓘ schemes over local fields ⓘ schemes over number fields ⓘ
basedOn	étale topology ⓘ
coefficientSystems	constructible sheaves ⓘ locally constant sheaves ⓘ ℓ-adic sheaves ⓘ
definedUsing	Grothendieck topology ⓘ Alexandrov–Čech cohomology ⓘ surface form: sheaf cohomology étale site ⓘ
developedBy	Alexander Grothendieck ⓘ
developedInContextOf	Weil conjectures ⓘ
field	algebraic geometry ⓘ arithmetic geometry ⓘ number theory ⓘ
formalizedIn	Séminaire de Géométrie Algébrique du Bois Marie ⓘ surface form: Séminaire de Géométrie Algébrique (SGA) Éléments de géométrie algébrique ⓘ
generalizes	singular cohomology ⓘ
hasComparisonIsomorphismWith	singular cohomology over complex numbers ⓘ
hasKeyConcept	Frobenius action on cohomology ⓘ constructible sheaf ⓘ trace formula ⓘ étale sheaf ⓘ ℓ-adic sheaf ⓘ
hasVariant	cohomology with supports ⓘ compactly supported étale cohomology ⓘ ℓ-adic étale cohomology ⓘ
introducedIn	1960s ⓘ
notionOfDegree	cohomological degree ⓘ
relatedTo	Betti cohomology ⓘ Galois cohomology ⓘ crystalline cohomology ⓘ de Rham cohomology ⓘ flat cohomology ⓘ
requires	category of schemes ⓘ homological algebra ⓘ sheaf theory ⓘ étale morphisms ⓘ
satisfies	Künneth formula under hypotheses ⓘ Mayer–Vietoris sequence in de Rham cohomology ⓘ surface form: Mayer–Vietoris sequence Poincaré duality for smooth proper varieties ⓘ long exact sequence of a pair ⓘ
typicalCoefficientRing	finite abelian group ⓘ ℓ-adic integers ⓘ ℚℓ ⓘ
usedFor	computing zeta functions of varieties ⓘ defining Chern classes ⓘ defining Galois representations ⓘ defining cycle class maps ⓘ defining ℓ-adic cohomology ⓘ proving the Weil conjectures ⓘ studying fundamental groups of schemes ⓘ studying schemes ⓘ studying torsion phenomena in algebraic geometry ⓘ studying varieties over finite fields ⓘ
usedIn	Langlands program ⓘ arithmetic of abelian varieties ⓘ arithmetic of elliptic curves ⓘ proof of Deligne’s theorem on Weil conjectures ⓘ study of motives ⓘ

Referenced by (3)

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

Alexander Grothendieck → knownFor → étale cohomology ⓘ

Séminaire de Géométrie Algébrique du Bois Marie → notableResult → étale cohomology ⓘ

this entity surface form: foundations of étale cohomology

Ramanujan–Petersson conjecture → proofUses → étale cohomology ⓘ