étale cohomology
E254118
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.
All labels observed (2)
| Label | Occurrences |
|---|---|
| étale cohomology canonical | 2 |
| foundations of étale cohomology | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2290621 — 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: étale cohomology Context triple: [Alexander Grothendieck, knownFor, étale cohomology]
-
A.
Weil cohomology
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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B.
Weil conjectures
The Weil conjectures are a set of deep statements about the zeta functions of algebraic varieties over finite fields that guided the development of modern algebraic geometry and were ultimately proved using étale cohomology.
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C.
Hodge theory
Hodge theory is a branch of mathematics that studies the relationship between differential forms, cohomology, and complex geometry, particularly on complex manifolds.
-
D.
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.
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E.
Adeles and Algebraic Groups
"Adeles and Algebraic Groups" is a foundational mathematical work by André Weil that develops the theory of adeles and its deep connections with algebraic groups and number theory.
- 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: étale cohomology Target entity description: 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.
-
A.
Weil cohomology
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.
-
B.
Weil conjectures
The Weil conjectures are a set of deep statements about the zeta functions of algebraic varieties over finite fields that guided the development of modern algebraic geometry and were ultimately proved using étale cohomology.
-
C.
Hodge theory
Hodge theory is a branch of mathematics that studies the relationship between differential forms, cohomology, and complex geometry, particularly on complex manifolds.
-
D.
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.
-
E.
Adeles and Algebraic Groups
"Adeles and Algebraic Groups" is a foundational mathematical work by André Weil that develops the theory of adeles and its deep connections with algebraic groups and number theory.
- F. None of above. chosen
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 ⓘ |
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: étale cohomology Description of subject: 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.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
foundations of étale cohomology