Weil cohomology

E244845

cohomology theory mathematical theory

Weil cohomology is a type of cohomology theory for algebraic varieties that satisfies specific axioms enabling the proof of the Weil conjectures and the development of modern algebraic geometry.

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All labels observed (1)

Label	Occurrences
Weil cohomology canonical	1

Statements (50)

Predicate	Object
instanceOf	cohomology theory ⓘ mathematical theory ⓘ
appliesTo	algebraic varieties ⓘ smooth projective varieties ⓘ varieties over a field ⓘ
associatedWith	André Weil ⓘ
assumes	resolution of singularities for some constructions ⓘ
codomain	finite-dimensional graded vector space ⓘ graded commutative algebra ⓘ
coefficientsIn	field of characteristic 0 ⓘ
definedOver	field of characteristic 0 ⓘ field of characteristic p ⓘ
developedInContextOf	Weil conjectures ⓘ
field	algebraic geometry ⓘ
goal	encode arithmetic and geometric information of varieties ⓘ
hasAxiom	Künneth formula ⓘ Lefschetz fixed-point theorem ⓘ surface form: Lefschetz trace formula Mayer–Vietoris sequence in de Rham cohomology ⓘ surface form: Mayer–Vietoris sequence Poincaré duality ⓘ cycle class map ⓘ excision ⓘ finite-dimensionality ⓘ functoriality ⓘ Hard Lefschetz theorem ⓘ surface form: hard Lefschetz theorem homotopy invariance ⓘ
hasExample	Betti cohomology ⓘ crystalline cohomology ⓘ de Rham cohomology ⓘ rigid cohomology ⓘ ℓ-adic étale cohomology ⓘ
hasStructure	Galois action on cohomology groups ⓘ graded-commutative cup product ⓘ intersection pairing ⓘ
implies	functional equation for zeta function under suitable conditions ⓘ rationality of zeta function of a smooth projective variety ⓘ
relatedTo	Hodge theory ⓘ Galois representations ⓘ surface form: Tate modules motivic cohomology ⓘ
requires	coefficient field of characteristic 0 for standard axioms ⓘ
satisfies	Künneth isomorphism for products of varieties ⓘ Lefschetz fixed-point theorem ⓘ surface form: Lefschetz fixed point formula Poincaré duality for smooth projective varieties ⓘ compatibility with cycle classes ⓘ hard Lefschetz isomorphisms ⓘ
usedFor	construction of numerical equivalence of cycles ⓘ definition of the Tate conjecture ⓘ definition of the standard conjectures on algebraic cycles ⓘ definition of zeta functions of varieties ⓘ proof of the Weil conjectures ⓘ study of algebraic cycles ⓘ

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Input

Subject: Weil cohomology
Description of subject: Weil cohomology is a type of cohomology theory for algebraic varieties that satisfies specific axioms enabling the proof of the Weil conjectures and the development of modern algebraic geometry.

Referenced by (1)

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

André Weil → notableConcept → Weil cohomology ⓘ