Herbrand quotient
Encyclopedia
In mathematics
, the Herbrand quotient is a quotient
of orders of cohomology
groups of a cyclic group
. It was invented by Jacques Herbrand
. It has an important application in class field theory
.
A, then the cohomology groups Hn(G,A) have period 2 for n≥1; in other words
an isomorphism induced by cup product
with a generator of H2(G,Z). (If instead we use the Tate cohomology group
s then the periodicity extends down to n=0.)
The Herbrand quotient h(G,A) is defined to be the quotient
of the order of the even and odd cohomology groups, if both are finite.
s of an Abelian group
, f and g, which satisfy the condition fg = gf = 0. Their Herbrand quotient q(f,g) is defined as
if the two indices are finite. If G is a cyclic group with generator γ acting on an Abelian group A, then we recover the previous definition by taking f = 1 - γ and g = 1 + γ + γ2 + ... .
is exact, then
These properties mean that the Herbrand quotient is usually relatively easy to calculate, and is often much easier to calculate than the orders of either of the individual cohomology groups.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Herbrand quotient is a quotient
Quotient
In mathematics, a quotient is the result of division. For example, when dividing 6 by 3, the quotient is 2, while 6 is called the dividend, and 3 the divisor. The quotient further is expressed as the number of times the divisor divides into the dividend e.g. The quotient of 6 and 2 is also 3.A...
of orders of cohomology
Cohomology
In mathematics, specifically in algebraic topology, cohomology is a general term for a sequence of abelian groups defined from a co-chain complex. That is, cohomology is defined as the abstract study of cochains, cocycles, and coboundaries...
groups of a cyclic group
Cyclic group
In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
. It was invented by Jacques Herbrand
Jacques Herbrand
Jacques Herbrand was a French mathematician who was born in Paris, France and died in La Bérarde, Isère, France. Although he died at only 23 years of age, he was already considered one of "the greatest mathematicians of the younger generation" by his professors Helmut Hasse, and Richard Courant.He...
. It has an important application in class field theory
Class field theory
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...
.
Definition
If G is a finite cyclic group acting on a G-moduleG-module
In mathematics, given a group G, a G-module is an abelian group M on which G acts compatibly with the abelian group structure on M. This widely applicable notion generalizes that of a representation of G...
A, then the cohomology groups Hn(G,A) have period 2 for n≥1; in other words
- Hn(G,A) = Hn+2(G,A),
an isomorphism induced by cup product
Cup product
In mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree p and q to form a composite cocycle of degree p + q. This defines an associative graded commutative product operation in cohomology, turning the cohomology of a space X into a...
with a generator of H2(G,Z). (If instead we use the Tate cohomology group
Tate cohomology group
In mathematics, Tate cohomology groups are a slightly modified form of the usual cohomology groups of a finite group that combine homology and cohomology groups into one sequence...
s then the periodicity extends down to n=0.)
The Herbrand quotient h(G,A) is defined to be the quotient
- h(G,A) = |H2(G,A)|/|H1(G,A)|
of the order of the even and odd cohomology groups, if both are finite.
Alternative definition
The quotient may be defined for a pair of endomorphismEndomorphism
In mathematics, an endomorphism is a morphism from a mathematical object to itself. For example, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about...
s of an Abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
, f and g, which satisfy the condition fg = gf = 0. Their Herbrand quotient q(f,g) is defined as
if the two indices are finite. If G is a cyclic group with generator γ acting on an Abelian group A, then we recover the previous definition by taking f = 1 - γ and g = 1 + γ + γ2 + ... .
Properties
- The Herbrand quotient is multiplicativeMultiplicative functionIn number theory, a multiplicative function is an arithmetic function f of the positive integer n with the property that f = 1 and whenevera and b are coprime, then...
on short exact sequences. In other words, if
- 0 → A → B → C → 0
is exact, then
- h(G,B) = h(G,A)h(G,C)
- If A is finite then h(G,A) = 1
- If Z is the integers with G acting trivially, then h(G,Z) = |G|
- If A is a finitely generated G-module, then the Herbrand quotient h(A) depends only on the complex G-module C⊗A (and so can be read off from the character of this complex representation of G).
These properties mean that the Herbrand quotient is usually relatively easy to calculate, and is often much easier to calculate than the orders of either of the individual cohomology groups.