The following situation—namely, the cohomology of induced objects—occurs very frequently, and we will devote a post to its analysis. Let ${G}$ be a cyclic group acting on an abelian group ${A}$. Suppose we have a decomposition ${A = \bigoplus_{i \in I} A_i}$ such that any two ${A_i}$ are isomorphic and ${G}$ permutes the ${A_i}$ with each other. It turns out that the computation of the cohomology of ${A}$ can often be simplified.

Then let ${G_0}$ be the stabilizer of ${A_{i_0}}$ for some fixed ${i_0 \in I}$, i.e. ${G_0 = \{g: gA_{i_0} = A_{i_0} \}.}$ Then, we have ${A = \mathrm{Ind}_{G_0}^G A_{i_0}}$. This is what I meant about ${A}$ being induced.

I claim that

$\displaystyle \boxed{ H_T^i(G, A) \simeq H_T^i(G_0, A_{i_0}) , \quad i = -1, 0. }$

In particular, we get an equality of the Herbrand quotients ${Q(A), Q(A_{i_0})}$. (more…)

We continue our quest to climb Mount Takagi-Artin.

In class field theory, it will be important to compute and keep track of the orders of groups such as ${(K^*:NL^*)}$, where ${L/K}$ is a Galois extension of local fields. A convenient piece of machinery for doing this is the Herbrand quotient, which we discuss today. I only sketch the proofs though, and a little familiarity with the Tate cohomology groups will be useful (but is not strictly necessary if one accepts the essentially combinatorial results without proof or proves them directly).

1. Definition

Let ${G}$ be a cyclic group generated by ${\sigma}$ and ${A}$ a ${G}$-module. It is well-known that the Tate cohomology groups ${H^i_T(G, A)}$ are periodic with period two and thus determined by ${H^0}$ and ${H^{-1}}$. By definition,

$\displaystyle H^0(G,A) = A^G/ NA ,$

where ${A^G}$ consists of the elements of ${A}$ fixed by ${G}$, and ${N: A \rightarrow A}$ is the norm map, ${a \rightarrow \sum_g ga}$. Moreover,

$\displaystyle H^{-1}(G, A) = \mathrm{ker} N/ (\sigma -1) A.$

(Normally, for ${G}$ only assumed finite, we would quotient by the sum of ${(\sigma - 1)A}$ for ${\sigma \in G}$ arbitrary, but here it is enough to do it for a generator—easy exercise.)

If both cohomology groups are finite, define the Herbrand quotient ${Q(A)}$ as

$\displaystyle Q(A) = \frac{ |H_T^0(G,A)|}{|H_T^{-1}(G,A)|}.$ (more…)