Herbrand quotient
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In mathematics, the Herbrand quotient is a quotient of orders of cohomology groups of a cyclic group. It was invented by Jacques Herbrand.
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[edit] Definition
If G is a finite cyclic group acting on a module A, then the cohomology groups Hn(G,A) have period 2 for n≥1; in other words
- Hn(G,A) = Hn+2(G,A).
The Herbrand quoteint 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.
[edit] Properties
- The Herbrand quotient is mutlipicative 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.
[edit] See also
[edit] References
The chapter by Atiyah and Wall in Algebraic Number Theory by J. W. S. Cassels, A. Frohlich ISBN 0-12-163251-2
