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

Then let be the stabilizer of for some fixed , i.e. Then, we have . This is what I meant about being induced.

I claim that

In particular, we get an equality of the Herbrand quotients . (more…)