Chow groups
E921616
Chow groups are algebraic invariants in algebraic geometry that classify algebraic cycles on a variety up to rational equivalence, playing a central role in the study of motives and intersection theory.
All labels observed (5)
| Label | Occurrences |
|---|---|
| Arakelov Chow group | 1 |
| Chow group of 0-cycles | 1 |
| Chow group of codimension k cycles | 1 |
| Chow groups canonical | 1 |
| Chow ring of projective space | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T11365401 — 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: Chow groups Context triple: [Deligne cohomology, relatedTo, Chow groups]
-
A.
Standard Conjectures on Algebraic Cycles
The Standard Conjectures on Algebraic Cycles are a set of deep, still unproven hypotheses in algebraic geometry that aim to provide a foundational theory of algebraic cycles and their cohomological properties, underpinning much of the modern theory of motives.
-
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.
-
C.
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.
-
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.
Serre’s cohomological methods in algebraic geometry
Serre’s cohomological methods in algebraic geometry are foundational techniques that use sheaf cohomology to relate and study algebraic and analytic geometry, profoundly influencing modern algebraic geometry and complex 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: Chow groups Target entity description: Chow groups are algebraic invariants in algebraic geometry that classify algebraic cycles on a variety up to rational equivalence, playing a central role in the study of motives and intersection theory.
-
A.
Standard Conjectures on Algebraic Cycles
The Standard Conjectures on Algebraic Cycles are a set of deep, still unproven hypotheses in algebraic geometry that aim to provide a foundational theory of algebraic cycles and their cohomological properties, underpinning much of the modern theory of motives.
-
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.
-
C.
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.
-
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.
Serre’s cohomological methods in algebraic geometry
Serre’s cohomological methods in algebraic geometry are foundational techniques that use sheaf cohomology to relate and study algebraic and analytic geometry, profoundly influencing modern algebraic geometry and complex geometry.
- F. None of above. chosen
Statements (66)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic invariant
ⓘ
cohomology theory ⓘ functor ⓘ |
| classify | algebraic cycles up to rational equivalence ⓘ |
| definedOn |
Deligne–Mumford stacks (via extensions)
ⓘ
algebraic varieties ⓘ schemes of finite type over a field ⓘ |
| dependsOn | choice of base field ⓘ |
| developedBy |
Pierre Deligne
ⓘ
Spencer Bloch ⓘ Steven Kleiman ⓘ William Fulton ⓘ Yuri Manin NERFINISHED ⓘ |
| field | algebraic geometry ⓘ |
| formalizedBy | Alexander Grothendieck ⓘ |
| functoriality |
contravariant for flat morphisms
ⓘ
contravariant for l.c.i. morphisms ⓘ covariant for proper morphisms ⓘ |
| generalizationOf |
Picard group of a smooth projective variety
ⓘ
divisor class group ⓘ |
| hasComponent |
Chow group of 0-cycles
ⓘ
linked to:
Chow groups
Chow group of 1-cycles ⓘ Chow group of codimension k cycles ⓘ
linked to:
Chow groups
Chow ring ⓘ |
| hasNotation |
A^i(X)
ⓘ
A_k(X) ⓘ CH^i(X) ⓘ CH_k(X) ⓘ |
| hasSpecialCase |
Chow group of 0-cycles on a smooth projective curve is its Picard group
ⓘ
Chow group of a point is isomorphic to Z ⓘ Chow ring of projective space is a polynomial ring modulo one relation ⓘ |
| hasStructure |
graded abelian group
ⓘ
ring (via intersection product) ⓘ |
| introducedBy | Wei-Liang Chow ⓘ |
| invariantUnder | rational equivalence of cycles ⓘ |
| namedAfter | Wei-Liang Chow ⓘ |
| notInvariantUnder | arbitrary birational maps in general ⓘ |
| relatedConcept |
Abel–Jacobi map
ⓘ
Bloch–Beilinson conjectures ⓘ
linked to:
Bloch’s conjecture on Chow groups
Chow motive ⓘ Chow–Künneth decomposition ⓘ Fulton’s intersection theory ⓘ
linked to:
Fulton’s Intersection Theory
Griffiths group ⓘ Grothendieck group of coherent sheaves ⓘ
linked to:
Grothendieck group
Hodge theory ⓘ K-theory of varieties ⓘ Néron–Severi group ⓘ Picard group ⓘ algebraic cycles ⓘ cycle class map ⓘ homological equivalence ⓘ motivic cohomology ⓘ numerical equivalence ⓘ rational equivalence ⓘ standard conjectures on algebraic cycles ⓘ étale cohomology ⓘ |
| satisfies |
base change formula
ⓘ
homotopy invariance for vector bundles ⓘ localization exact sequence ⓘ projection formula ⓘ |
| usedFor |
enumerative geometry
ⓘ
formulating conjectures in arithmetic geometry ⓘ formulating cycle class maps ⓘ intersection theory ⓘ studying birational invariants ⓘ theory of motives ⓘ |
How these facts were elicited
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Instruction
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Input
Subject: Chow groups Description of subject: Chow groups are algebraic invariants in algebraic geometry that classify algebraic cycles on a variety up to rational equivalence, playing a central role in the study of motives and intersection theory.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: Chow groups
linked to: Chow groups
linked to: Chow groups
linked to: Chow groups