Herbrand quotient
E839565
The Herbrand quotient is an invariant in algebraic number theory and group cohomology that measures the relative sizes of certain cohomology groups associated with a finite group action on a module.
Statements (44)
| Predicate | Object |
|---|---|
| instanceOf |
algebraic number theory concept
ⓘ
cohomological invariant ⓘ group cohomology concept ⓘ |
| appearsIn |
cohomological formulas for class groups
ⓘ
cohomological formulas for unit groups ⓘ |
| appliesTo |
G-modules
ⓘ
finite group actions ⓘ |
| associatedWith |
H^0(G,M)
ⓘ
H^1(G,M) ⓘ |
| assumes |
G finite
ⓘ
M a finitely generated G-module in arithmetic applications ⓘ |
| category | mathematical invariant ⓘ |
| context | finite Galois extensions of number fields ⓘ |
| definedBy | Jacques Herbrand in the context of class field theory NERFINISHED ⓘ |
| definedFor |
G-module M
ⓘ
finite group G ⓘ |
| domain | cohomology of groups ⓘ |
| field |
algebraic number theory
ⓘ
group cohomology ⓘ |
| formalism | group cohomology ⓘ |
| generalizes | index computations in cohomology ⓘ |
| hasFormula | h(G,M) = |H^0(G,M)| / |H^1(G,M)| when both groups are finite ⓘ |
| hasProperty |
equals 1 for many natural G-modules in number theory
ⓘ
multiplicative in short exact sequences of G-modules under finiteness conditions ⓘ |
| invariantOf | pair (G,M) ⓘ |
| invariantUnder | isomorphisms of G-modules ⓘ |
| measures | relative sizes of cohomology groups ⓘ |
| namedAfter | Jacques Herbrand NERFINISHED ⓘ |
| relatedTo |
Euler characteristic in group cohomology
ⓘ
Herbrand’s theorem NERFINISHED ⓘ Tate cohomology groups ⓘ |
| requires |
H^0(G,M) finite
ⓘ
H^1(G,M) finite ⓘ |
| toolFor | relating arithmetic invariants to cohomology groups ⓘ |
| usedIn |
Galois cohomology
NERFINISHED
ⓘ
Galois module structure of units ⓘ Tate cohomology theory NERFINISHED ⓘ class field theory NERFINISHED ⓘ proofs of relations between unit ranks and class numbers ⓘ study of ideal class groups ⓘ |
| usedToCompare | fixed points and coinvariants of G on M ⓘ |
| usedToStudy |
Galois action on ideal class groups
ⓘ
Galois action on units of number fields ⓘ |
| valuesIn | positive rational numbers ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.