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 (2)

Label	Occurrences
Weil cohomology theory	2
Weil cohomology canonical	1

How this entity was disambiguated

This entity first appeared as the object of triple T2228055 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: Weil cohomology
Context triple: [André Weil, notableConcept, Weil cohomology]

A. Hodge theory
Hodge theory is a branch of mathematics that studies the relationship between differential forms, cohomology, and complex geometry, particularly on complex manifolds.
B. Chern classes
Chern classes are fundamental topological invariants in differential and algebraic geometry that classify complex vector bundles and capture their curvature and twisting properties.
C. Hodge Conjecture
The Hodge Conjecture is a major unsolved problem in algebraic geometry that predicts which cohomology classes on a non-singular projective complex variety arise from algebraic subvarieties.
D. Hasse–Weil zeta function
The Hasse–Weil zeta function is an analytic object in number theory that encodes arithmetic information about algebraic varieties over number fields, generalizing the Riemann zeta function and playing a central role in modern arithmetic geometry and conjectures like the Weil conjectures and the Birch–Swinnerton-Dyer conjecture.
E. Alexandrov–Čech cohomology
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.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: Weil cohomology
Target entity description: 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.

A. Hodge theory
Hodge theory is a branch of mathematics that studies the relationship between differential forms, cohomology, and complex geometry, particularly on complex manifolds.
B. Chern classes
Chern classes are fundamental topological invariants in differential and algebraic geometry that classify complex vector bundles and capture their curvature and twisting properties.
C. Hodge Conjecture
The Hodge Conjecture is a major unsolved problem in algebraic geometry that predicts which cohomology classes on a non-singular projective complex variety arise from algebraic subvarieties.
D. Hasse–Weil zeta function
The Hasse–Weil zeta function is an analytic object in number theory that encodes arithmetic information about algebraic varieties over number fields, generalizing the Riemann zeta function and playing a central role in modern arithmetic geometry and conjectures like the Weil conjectures and the Birch–Swinnerton-Dyer conjecture.
E. Alexandrov–Čech cohomology
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.
F. None of above. chosen

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 trace formula ⓘ linked to: Lefschetz fixed-point theorem Mayer–Vietoris sequence ⓘ linked to: Mayer–Vietoris sequence in de Rham cohomology Poincaré duality ⓘ cycle class map ⓘ excision ⓘ finite-dimensionality ⓘ functoriality ⓘ hard Lefschetz theorem ⓘ linked to: 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 ⓘ Tate modules ⓘ linked to: Galois representations motivic cohomology ⓘ
requires	coefficient field of characteristic 0 for standard axioms ⓘ
satisfies	Künneth isomorphism for products of varieties ⓘ Lefschetz fixed point formula ⓘ linked to: Lefschetz fixed-point theorem 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 ⓘ

How these facts were elicited

Referenced by (3)

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

André Weil → notableConcept → Weil cohomology ⓘ

Standard Conjectures on Algebraic Cycles → relatedTo → Weil cohomology theory ⓘ

linked to: Weil cohomology

Grothendieck–Lefschetz trace formula → involves → Weil cohomology theory ⓘ