Let $K$ be a number field, $p > 3$ a prime and $\mathfrak{p}$ a prime of $K$ lying over $p$ and $n \geq 1$. Let $C$ be a subgroup of $(\mathcal{O}_K/\mathfrak{p}^n)^\times$ and consider $\mathcal{O}_K/\mathfrak{p}^n$ as a $C$-module via multiplication. Let $C'$ be the prime-to-$p$ part of $C$.
How can I prove that $H^i(C',\mathcal{O}_K/\mathfrak{p}^n) = 0$ for all $i$, arguing directly from the definitions (i.e with no high-powered (co)homological tools)?
Cheers!
The key point is corollary 1 of chapter 8 of the French edition (I suppose that the numbering is the same in the English edition). I recall that you wish to kill the cohomology of a finite group G acting on a finite module M. Your desire will be fulfilled by 2 facts:
-first, it follows trivially from the definition of cohomology via cocycles that for all $i\ge 1$, the groups $H^{i}$ are killed by the order of M
-second, using properties of the (co)restriction maps, the aforementioned corollary states that the $H^{i}$ are killed by the order of G
NB: we are dealing here with the ordinary cohomology groups, hence the requirement that $i\ge 1$. Since G is finite, we could as well replace them by the modified Tate cohomology groups and take all $i\in\mathbf Z$ ./.