Weil cohomology
E244845
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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| 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.
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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 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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
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.