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.
Statements (66)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic invariant
ⓘ
cohomology theory ⓘ functor ⓘ |
| classify | algebraic cycles up to rational equivalence ⓘ |
| definedOn |
Deligne–Mumford stacks (via extensions)
NERFINISHED
ⓘ
algebraic varieties ⓘ schemes of finite type over a field ⓘ |
| dependsOn | choice of base field ⓘ |
| developedBy |
Pierre Deligne
NERFINISHED
ⓘ
Spencer Bloch NERFINISHED ⓘ Steven Kleiman NERFINISHED ⓘ William Fulton NERFINISHED ⓘ Yuri Manin NERFINISHED ⓘ |
| field | algebraic geometry ⓘ |
| formalizedBy | Alexander Grothendieck NERFINISHED ⓘ |
| 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
NERFINISHED
ⓘ
Chow group of 1-cycles ⓘ Chow group of codimension k cycles NERFINISHED ⓘ 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 NERFINISHED ⓘ |
| invariantUnder | rational equivalence of cycles ⓘ |
| namedAfter | Wei-Liang Chow NERFINISHED ⓘ |
| notInvariantUnder | arbitrary birational maps in general ⓘ |
| relatedConcept |
Abel–Jacobi map
NERFINISHED
ⓘ
Bloch–Beilinson conjectures NERFINISHED ⓘ Chow motive ⓘ Chow–Künneth decomposition NERFINISHED ⓘ Fulton’s intersection theory NERFINISHED ⓘ Griffiths group NERFINISHED ⓘ Grothendieck group of coherent sheaves NERFINISHED ⓘ Hodge theory ⓘ K-theory of varieties ⓘ Néron–Severi group NERFINISHED ⓘ Picard group NERFINISHED ⓘ 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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.