Verdier duality
E620670
duality theory
mathematical concept
theorem in algebraic topology
theorem in homological algebra
theorem in sheaf theory
Verdier duality is a powerful generalization of Poincaré duality formulated in the language of derived categories and sheaf theory, providing a duality functor that relates cohomology with compact support to ordinary cohomology on possibly singular or non-compact spaces.
All labels observed (2)
| Label | Occurrences |
|---|---|
| Verdier duality canonical | 2 |
| theory of perverse sheaves | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6801413 — 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: Verdier duality Context triple: [Poincaré duality, generalizedBy, Verdier duality]
-
A.
Grothendieck duality
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.
-
B.
Serre duality
Serre duality is a fundamental theorem in algebraic geometry that generalizes classical duality for Riemann surfaces to higher-dimensional projective varieties, relating cohomology groups of coherent sheaves via a dualizing sheaf.
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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.
-
D.
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.
-
E.
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.
- 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: Verdier duality Target entity description: Verdier duality is a powerful generalization of Poincaré duality formulated in the language of derived categories and sheaf theory, providing a duality functor that relates cohomology with compact support to ordinary cohomology on possibly singular or non-compact spaces.
-
A.
Grothendieck duality
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.
-
B.
Serre duality
Serre duality is a fundamental theorem in algebraic geometry that generalizes classical duality for Riemann surfaces to higher-dimensional projective varieties, relating cohomology groups of coherent sheaves via a dualizing sheaf.
-
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.
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.
-
E.
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.
- F. None of above. chosen
Statements (51)
| Predicate | Object |
|---|---|
| instanceOf |
duality theory
ⓘ
mathematical concept ⓘ theorem in algebraic topology ⓘ theorem in homological algebra ⓘ theorem in sheaf theory ⓘ |
| appliesTo |
complex algebraic varieties
ⓘ
locally compact topological spaces ⓘ non-compact spaces ⓘ possibly singular spaces ⓘ schemes ⓘ |
| compatibleWith | six operations formalism ⓘ |
| defines |
Verdier duality functor
NERFINISHED
ⓘ
duality functor ⓘ |
| developedIn | 20th century ⓘ |
| expressesAsIsomorphism |
H_c^i(X, F) ≅ H^{-i}(X, D_X F)^∨ under finiteness conditions
ⓘ
RHom(F, D_X G) ≅ RHom(Rf_! F, G) ⓘ |
| field |
algebraic geometry
ⓘ
algebraic topology ⓘ derived category theory ⓘ homological algebra ⓘ sheaf theory ⓘ |
| formalizedIn |
language of derived categories of sheaves
ⓘ
language of triangulated categories ⓘ |
| frameworkFor |
Grothendieck’s six functors formalism
NERFINISHED
ⓘ
intersection cohomology ⓘ perverse sheaves ⓘ |
| generalizes | Poincaré duality NERFINISHED ⓘ |
| hasConsequence |
Poincaré duality for smooth manifolds
ⓘ
self-duality of intersection cohomology ⓘ |
| hasKeyFunctor |
derived direct image Rf_*
ⓘ
derived direct image with proper support Rf_! ⓘ derived functor RHom ⓘ extraordinary pullback f^! ⓘ |
| hasKeyObject |
Verdier dualizing complex
ⓘ
dualizing complex ⓘ |
| hasVariant |
Verdier duality for perverse sheaves
ⓘ
equivariant Verdier duality ⓘ relative Verdier duality NERFINISHED ⓘ |
| implies | Poincaré duality for compact oriented manifolds ⓘ |
| namedAfter | Jean-Louis Verdier NERFINISHED ⓘ |
| relatedTo |
Grothendieck duality theory
NERFINISHED
ⓘ
Serre duality NERFINISHED ⓘ |
| relates |
cohomology with compact support
ⓘ
ordinary cohomology ⓘ |
| usesConcept |
Grothendieck duality
NERFINISHED
ⓘ
bounded derived category of sheaves ⓘ cohomology with compact support ⓘ constructible sheaf ⓘ derived category ⓘ extraordinary inverse image functor ⓘ sheaf cohomology ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Verdier duality Description of subject: Verdier duality is a powerful generalization of Poincaré duality formulated in the language of derived categories and sheaf theory, providing a duality functor that relates cohomology with compact support to ordinary cohomology on possibly singular or non-compact spaces.
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
theory of perverse sheaves