Serre duality
E253115
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.
All labels observed (8)
How this entity was disambiguated
This entity first appeared as the object of triple T2306392 — 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: Serre duality Context triple: [Jean-Pierre Serre, notableWork, Serre duality]
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A.
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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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.
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C.
Riemann–Roch theorem
The Riemann–Roch theorem is a fundamental result in algebraic geometry and complex analysis that relates the dimension of spaces of meromorphic sections of a line bundle on a curve to topological data such as genus and degree.
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D.
Hodge theory
Hodge theory is a branch of mathematics that studies the relationship between differential forms, cohomology, and complex geometry, particularly on complex manifolds.
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E.
Chern classes
Chern classes are fundamental topological invariants in differential and algebraic geometry that classify complex vector bundles and capture their curvature and twisting properties.
- 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: Serre duality Target entity description: 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.
-
A.
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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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.
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C.
Riemann–Roch theorem
The Riemann–Roch theorem is a fundamental result in algebraic geometry and complex analysis that relates the dimension of spaces of meromorphic sections of a line bundle on a curve to topological data such as genus and degree.
-
D.
Hodge theory
Hodge theory is a branch of mathematics that studies the relationship between differential forms, cohomology, and complex geometry, particularly on complex manifolds.
-
E.
Chern classes
Chern classes are fundamental topological invariants in differential and algebraic geometry that classify complex vector bundles and capture their curvature and twisting properties.
- F. None of above. chosen
Statements (48)
| Predicate | Object |
|---|---|
| instanceOf |
duality theorem
ⓘ
theorem in algebraic geometry ⓘ |
| appearsIn |
EGA III
ⓘ
Hartshorne Algebraic Geometry ⓘ |
| appliesTo |
coherent sheaves
ⓘ
projective varieties ⓘ proper schemes ⓘ |
| assumes | base field often algebraically closed ⓘ |
| context |
coherent cohomology
ⓘ
derived categories of coherent sheaves ⓘ sheaf cohomology ⓘ |
| dimensionCondition | n = dim X ⓘ |
| field | algebraic geometry ⓘ |
| formulation | Ext^i(F,ω_X) is dual to H^{n-i}(X,F) ⓘ |
| generalizes |
Poincaré duality for Riemann surfaces
ⓘ
classical duality for Riemann surfaces ⓘ |
| hasVariant |
Serre duality
self-linksurface differs
ⓘ
surface form:
Serre duality for curves
Serre duality self-linksurface differs ⓘ
surface form:
Serre duality for higher-dimensional varieties
Serre duality self-linksurface differs ⓘ
surface form:
Serre duality for surfaces
|
| historicalPeriod | mid 20th century ⓘ |
| implies |
finiteness of cohomology for coherent sheaves on proper schemes
ⓘ
symmetry properties of H^i and H^{n-i} ⓘ |
| involves |
Grothendieck duality
ⓘ
surface form:
Grothendieck duality theory
canonical line bundle ⓘ dualizing sheaf ⓘ |
| isSpecialCaseOf |
Grothendieck duality
ⓘ
surface form:
Grothendieck–Verdier duality
|
| mathematicsSubjectClassification |
14F05
ⓘ
14F17 ⓘ |
| namedAfter | Jean-Pierre Serre ⓘ |
| pairingType | perfect pairing of finite-dimensional k-vector spaces ⓘ |
| relatedConcept |
Serre duality
self-linksurface differs
ⓘ
surface form:
Serre functor
Serre vanishing theorem ⓘ canonical divisor ⓘ dualizing complex ⓘ |
| relatedTo |
Hodge theory on algebraic varieties
ⓘ
Riemann–Roch theorem ⓘ |
| relates |
Ext-groups and cohomology groups
ⓘ
cohomology groups of coherent sheaves ⓘ global sections and top-degree cohomology ⓘ |
| requires |
finite-dimensional cohomology groups over the base field
ⓘ
properness of the variety or scheme ⓘ |
| states |
Serre duality
self-linksurface differs
ⓘ
surface form:
for a smooth projective variety X over a field k and a coherent sheaf F on X, H^i(X,F) is dual to H^{n-i}(X,F^∨ ⊗ ω_X)
|
| usedFor |
Riemann–Roch theorem
ⓘ
surface form:
Riemann–Roch type formulas
classification of line bundles on curves ⓘ computing dimensions of cohomology groups ⓘ proving vanishing theorems ⓘ study of canonical models of varieties ⓘ |
| uses |
Serre duality
self-linksurface differs
ⓘ
surface form:
Serre duality pairing
|
How these facts were elicited
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Instruction
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Input
Subject: Serre duality Description of subject: 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.
Referenced by (10)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
Serre duality (in modern formulations)
this entity surface form:
for a smooth projective variety X over a field k and a coherent sheaf F on X, H^i(X,F) is dual to H^{n-i}(X,F^∨ ⊗ ω_X)
this entity surface form:
Serre duality pairing
this entity surface form:
Serre duality for curves
this entity surface form:
Serre duality for surfaces
this entity surface form:
Serre duality for higher-dimensional varieties
this entity surface form:
Serre functor