Grothendieck duality
E254134
Grothendieck duality is a foundational theory in algebraic geometry that generalizes classical Serre duality to a broad categorical and sheaf-theoretic framework for studying duality on schemes and morphisms.
All labels observed (3)
| Label | Occurrences |
|---|---|
| Grothendieck duality canonical | 1 |
| Grothendieck duality theory | 1 |
| Grothendieck–Verdier duality | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T2290655 — 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: Grothendieck duality Context triple: [Alexander Grothendieck, notableConcept, Grothendieck duality]
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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.
Sheaves in Geometry and Logic
Sheaves in Geometry and Logic is a foundational monograph that develops the theory of sheaves and topos theory and explores their deep connections to geometry, logic, and the foundations of mathematics.
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C.
Poincaré duality
Poincaré duality is a fundamental theorem in algebraic topology that relates the homology and cohomology groups of an oriented closed manifold in complementary dimensions.
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D.
Categories for the Working Mathematician
Categories for the Working Mathematician is a foundational textbook in category theory that systematically develops the subject and its applications for professional mathematicians.
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E.
Hodge theory
Hodge theory is a branch of mathematics that studies the relationship between differential forms, cohomology, and complex geometry, particularly on complex manifolds.
- 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: Grothendieck duality Target entity description: Grothendieck duality is a foundational theory in algebraic geometry that generalizes classical Serre duality to a broad categorical and sheaf-theoretic framework for studying duality on schemes and morphisms.
-
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.
Sheaves in Geometry and Logic
Sheaves in Geometry and Logic is a foundational monograph that develops the theory of sheaves and topos theory and explores their deep connections to geometry, logic, and the foundations of mathematics.
-
C.
Poincaré duality
Poincaré duality is a fundamental theorem in algebraic topology that relates the homology and cohomology groups of an oriented closed manifold in complementary dimensions.
-
D.
Categories for the Working Mathematician
Categories for the Working Mathematician is a foundational textbook in category theory that systematically develops the subject and its applications for professional mathematicians.
-
E.
Hodge theory
Hodge theory is a branch of mathematics that studies the relationship between differential forms, cohomology, and complex geometry, particularly on complex manifolds.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
duality theory
ⓘ
theory in algebraic geometry ⓘ |
| appliesTo |
finite morphisms of schemes
ⓘ
morphisms of schemes ⓘ proper morphisms of schemes ⓘ schemes ⓘ smooth morphisms of schemes ⓘ |
| centralConcept |
adjoint functors
ⓘ
derived categories ⓘ dualizing complex ⓘ extraordinary inverse image functor ⓘ |
| context |
coherent sheaves
ⓘ
finite type morphisms ⓘ proper morphisms of schemes ⓘ quasi-coherent sheaves ⓘ |
| developedBy | Alexander Grothendieck ⓘ |
| developedIn |
EGA
ⓘ
SGA ⓘ |
| field | algebraic geometry ⓘ |
| formalism | six functors formalism ⓘ |
| frameworkType |
categorical
ⓘ
sheaf-theoretic ⓘ |
| furtherDevelopedBy |
Amnon Neeman
ⓘ
Brian Conrad ⓘ Joseph Lipman ⓘ Robin Hartshorne ⓘ |
| generalizes | Serre duality ⓘ |
| goal |
to express cohomology in terms of dual objects
ⓘ
to generalize classical duality on varieties ⓘ |
| hasAspect |
absolute duality
ⓘ
relative duality ⓘ |
| hasGeneralization | non-noetherian duality theories ⓘ |
| involves |
base change theorems
ⓘ
bounded derived category of coherent sheaves ⓘ perfect complexes ⓘ trace morphisms ⓘ |
| namedAfter | Alexander Grothendieck ⓘ |
| provides |
global duality statements
ⓘ
local duality statements ⓘ relative duality for morphisms ⓘ |
| relatedConcept |
Serre duality
ⓘ
Verdier duality ⓘ canonical sheaf ⓘ dualizing sheaf ⓘ |
| requires | noetherian hypotheses in classical form ⓘ |
| usedIn |
birational geometry
ⓘ
intersection theory ⓘ moduli theory ⓘ |
| usesFunctor |
derived direct image functor Rf_*
ⓘ
extraordinary inverse image functor f^! ⓘ |
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: Grothendieck duality Description of subject: Grothendieck duality is a foundational theory in algebraic geometry that generalizes classical Serre duality to a broad categorical and sheaf-theoretic framework for studying duality on schemes and morphisms.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Grothendieck duality theory
this entity surface form:
Grothendieck–Verdier duality