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.
Observed surface forms (1)
| Surface form | Occurrences |
|---|---|
| theory of perverse sheaves | 1 |
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 ⓘ |
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