If G is a finite cyclic group acting on a G-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),
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) = |H2(G,A)|/|H1(G,A)|
of the order of the even and odd cohomology groups.
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 + ... .
- 0 → A → B → C → 0
- h(G,B) = h(G,A)h(G,C)
- If A is finite then h(G,A) = 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,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.
- Cohen (2007) p.245
- Serre (1979) p.134
- 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.