- Herbrand quotient
In mathematics, the Herbrand quotient is a
quotient of orders ofcohomology groups of acyclic group. It was invented byJacques Herbrand .Definition
If "G" is a finite cyclic group acting on a 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").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, if both are finite.Properties
*The Herbrand quotient is
multiplicative on short exact sequences. In other words, if:0 → "A" → "B" → "C" → 0is 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.ee also
*
Class formation References
The chapter by Atiyah and Wall in "Algebraic Number Theory" by J. W. S. Cassels, A. Frohlich ISBN 0-12-163251-2
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