Herbrand quotient


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.

Definition

If G is a finite cyclic group acting on a G-module A, then the cohomology groups Hn have period 2 for n≥1; in other words
an isomorphism induced by cup product with a generator of H2.
A Herbrand module is an A for which the cohomology groups are finite. In this case, the Herbrand quotient h is defined to be the quotient
of the order of the even and odd cohomology groups.

Alternative definition

The quotient may be defined for a pair of endomorphisms of an Abelian group, f and g, which satisfy the condition fg = gf = 0. Their Herbrand quotient q 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

is exact, and any two of the quotients are defined, then so is the third and
  • If A is finite then h = 1.
  • For A is a submodule of the G-module B of finite index, if either quotient is defined then so is the other and they are equal: more generally, if there is a G-morphism A → B with finite kernel and cokernel then the same holds.
  • If Z is the integers with G acting trivially, then h = |G|
  • If A is a finitely generated G-module, then the Herbrand quotient h depends only on the complex G-module C⊗A.
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.