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…)