Tate cohomology
E883484
Tate cohomology is a refinement of group cohomology that extends cohomology and homology to negative degrees, playing a key role in algebraic number theory and Galois cohomology.
Statements (49)
| Predicate | Object |
|---|---|
| instanceOf |
cohomology theory
ⓘ
construction in homological algebra ⓘ mathematical concept ⓘ |
| appearsIn |
algebraic K-theory computations
ⓘ
equivariant stable homotopy theory ⓘ |
| appliesTo |
Galois groups
NERFINISHED
ⓘ
finite groups ⓘ |
| context |
Galois representations
ⓘ
representation theory of finite groups ⓘ |
| definedBy | splicing projective resolutions in positive and negative degrees ⓘ |
| definedOver | group algebra of G over a ring ⓘ |
| domain | group G and G-module M ⓘ |
| extends |
group cohomology
ⓘ
group homology ⓘ |
| field |
Galois cohomology
NERFINISHED
ⓘ
algebra ⓘ algebraic number theory ⓘ homological algebra ⓘ |
| generalizes | Herbrand cohomology NERFINISHED ⓘ |
| hasFeature |
built from complete resolutions
ⓘ
coincides with group cohomology in positive degrees ⓘ coincides with group homology in negative degrees up to shift ⓘ defined in all integer degrees ⓘ includes negative cohomological degrees ⓘ |
| hasNotation | \hat{H}^n(G,M) ⓘ |
| hasProperty |
compatible with restriction and corestriction maps
ⓘ
functorial in the module argument ⓘ periodic for finite cyclic groups ⓘ |
| hasVariant |
Tate cohomology for profinite groups
ⓘ
Tate–Farrell cohomology NERFINISHED ⓘ |
| namedAfter | John Tate NERFINISHED ⓘ |
| refines | group cohomology ⓘ |
| relatedTo |
Herbrand quotient
NERFINISHED
ⓘ
Poitou–Tate duality NERFINISHED ⓘ Tate duality NERFINISHED ⓘ class field theory ⓘ global class field theory ⓘ local class field theory ⓘ |
| satisfies |
dimension shifting properties
ⓘ
long exact sequences ⓘ |
| usedFor |
Galois module structure analysis
ⓘ
duality theorems in number theory ⓘ study of ideal class groups ⓘ study of units in number fields ⓘ |
| usesConcept |
Ext functor
ⓘ
Tor functor ⓘ complete resolution ⓘ group ring ⓘ projective resolution ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.