# 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 *H*^{n}(*G*,*A*) have period 2 for *n*≥1; in other words

*H*^{n}(*G*,*A*) =*H*^{n+2}(*G*,*A*),

an isomorphism induced by cup product with a generator of *H*^{2}(*G*,**Z**). (If instead we use the Tate cohomology groups then the periodicity extends down to *n*=0.)

A **Herbrand module** is an *A* for which the cohomology groups are finite. In this case, the **Herbrand quotient** *h*(*G*,*A*) is defined to be the quotient

*h*(*G*,*A*) = |*H*^{2}(*G*,*A*)|/|*H*^{1}(*G*,*A*)|

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*(*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 + γ^{2} + ... .

## Properties

- The Herbrand quotient is multiplicative on short exact sequences.[1] In other words, if

- 0 →
*A*→*B*→*C*→ 0

is exact, and any two of the quotients are defined, then so is the third and[2]

*h*(*G*,*B*) =*h*(*G*,*A*)*h*(*G*,*C*)

- If
*A*is finite then*h*(*G*,*A*) = 1.[2] - 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:[1] more generally, if there is a*G*-morphism*A*→*B*with finite kernel and cokernel then the same holds.[2] - 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.

## See also

## Notes

- Cohen (2007) p.245
- Serre (1979) p.134

## References

- Atiyah, M.F.; Wall, C.T.C. (1967). "Cohomology of Groups". In Cassels, J.W.S.; Fröhlich, Albrecht (eds.).
*Algebraic Number Theory*. Academic Press. Zbl 0153.07403. See section 8. - Artin, Emil; Tate, John (2009).
*Class Field Theory*. AMS Chelsea. p. 5. ISBN 0-8218-4426-1. Zbl 1179.11040. - Cohen, Henri (2007).
*Number Theory – Volume I: Tools and Diophantine Equations*. Graduate Texts in Mathematics.**239**. Springer-Verlag. pp. 242–248. ISBN 978-0-387-49922-2. Zbl 1119.11001. - Janusz, Gerald J. (1973).
*Algebraic number fields*. Pure and Applied Mathematics.**55**. Academic Press. p. 142. Zbl 0307.12001. - Koch, Helmut (1997).
*Algebraic Number Theory*. Encycl. Math. Sci.**62**(2nd printing of 1st ed.). Springer-Verlag. pp. 120–121. ISBN 3-540-63003-1. Zbl 0819.11044. - Serre, Jean-Pierre (1979).
*Local Fields*. Graduate Texts in Mathematics.**67**. Translated by Greenberg, Marvin Jay. Springer-Verlag. ISBN 0-387-90424-7. Zbl 0423.12016.