Tate cohomology
E883484
Tate cohomology is a refinement of group cohomology that extends cohomology and homology to negative degrees, playing a key role in algebraic number theory and Galois cohomology.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Tate cohomology canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T10732933 — 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: Tate cohomology Context triple: [Cohomologie Galoisienne, topic, Tate cohomology]
-
A.
Deligne cohomology
Deligne cohomology is a refined cohomology theory in algebraic geometry that combines singular cohomology and differential forms to capture both topological and arithmetic information about complex algebraic varieties.
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B.
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.
-
C.
Alexander–Spanier cohomology
Alexander–Spanier cohomology is a cohomology theory in algebraic topology defined using cochains on all finite subsets of a space, notable for its generality and close relationship to Čech and singular cohomology.
-
D.
étale cohomology
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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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: Tate cohomology Target entity description: Tate cohomology is a refinement of group cohomology that extends cohomology and homology to negative degrees, playing a key role in algebraic number theory and Galois cohomology.
-
A.
Deligne cohomology
Deligne cohomology is a refined cohomology theory in algebraic geometry that combines singular cohomology and differential forms to capture both topological and arithmetic information about complex algebraic varieties.
-
B.
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.
-
C.
Alexander–Spanier cohomology
Alexander–Spanier cohomology is a cohomology theory in algebraic topology defined using cochains on all finite subsets of a space, notable for its generality and close relationship to Čech and singular cohomology.
-
D.
étale cohomology
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.
-
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 (49)
| Predicate | Object |
|---|---|
| instanceOf |
cohomology theory
ⓘ
construction in homological algebra ⓘ mathematical concept ⓘ |
| appearsIn |
algebraic K-theory computations
ⓘ
equivariant stable homotopy theory ⓘ |
| appliesTo |
Galois groups
NERFINISHED
ⓘ
finite groups ⓘ |
| context |
Galois representations
ⓘ
representation theory of finite groups ⓘ |
| definedBy | splicing projective resolutions in positive and negative degrees ⓘ |
| definedOver | group algebra of G over a ring ⓘ |
| domain | group G and G-module M ⓘ |
| extends |
group cohomology
ⓘ
group homology ⓘ |
| field |
Galois cohomology
NERFINISHED
ⓘ
algebra ⓘ algebraic number theory ⓘ homological algebra ⓘ |
| generalizes | Herbrand cohomology NERFINISHED ⓘ |
| hasFeature |
built from complete resolutions
ⓘ
coincides with group cohomology in positive degrees ⓘ coincides with group homology in negative degrees up to shift ⓘ defined in all integer degrees ⓘ includes negative cohomological degrees ⓘ |
| hasNotation | \hat{H}^n(G,M) ⓘ |
| hasProperty |
compatible with restriction and corestriction maps
ⓘ
functorial in the module argument ⓘ periodic for finite cyclic groups ⓘ |
| hasVariant |
Tate cohomology for profinite groups
ⓘ
Tate–Farrell cohomology NERFINISHED ⓘ |
| namedAfter | John Tate NERFINISHED ⓘ |
| refines | group cohomology ⓘ |
| relatedTo |
Herbrand quotient
NERFINISHED
ⓘ
Poitou–Tate duality NERFINISHED ⓘ Tate duality NERFINISHED ⓘ class field theory ⓘ global class field theory ⓘ local class field theory ⓘ |
| satisfies |
dimension shifting properties
ⓘ
long exact sequences ⓘ |
| usedFor |
Galois module structure analysis
ⓘ
duality theorems in number theory ⓘ study of ideal class groups ⓘ study of units in number fields ⓘ |
| usesConcept |
Ext functor
ⓘ
Tor functor ⓘ complete resolution ⓘ group ring ⓘ projective resolution ⓘ |
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: Tate cohomology Description of subject: Tate cohomology is a refinement of group cohomology that extends cohomology and homology to negative degrees, playing a key role in algebraic number theory and Galois cohomology.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.