Grothendieck duality

E254134

duality theory theory in algebraic geometry

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.

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All labels observed (4)

Label	Occurrences
Grothendieck duality theory	4
Grothendieck duality canonical	3
Grothendieck–Verdier duality	1
Grothendieck’s six functors formalism	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]

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
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