I am trying to learn some group cohomology and I'm starting to get my head around the theory, but I find it hard to find some explicit examples of the calculation of group cohomology of some small finite groups. For example, I think it would help my understanding to see a calculation of the Tate cohomology groups of $\mathbb{Z}/2\mathbb{Z}$ over $\mathbb{Z}$.
On wikipedia it says that $$\hat{H}^p(\mathbb{Z}/2\mathbb{Z};\mathbb{Z})=\begin{cases} 0, &p\text{ odd;}\\ \mathbb{Z}/2\mathbb{Z}, &p\text{ even}.\end{cases}$$
Could someone show me how to compute this by explicit calculation?
Thanks!
The action of any group G (here G = Z/2Z) on Z is always implicitly supposed to be the trivial action. If G is cyclic, there is a theorem saying that Tate cohomology is periodic, with period 2, more precisely H^p(G,M) is isomorphic to $H^{p+2}(G,M)$, so it suffices to compute H^0 and H^1, which are given explicitly by $H^0(G,M)=M^G/N(M)$ and $H^1(G,M)=KerN/(s-1)M$. Here s denotes a generator of G and N, the norm (or trace) map, is defined by $N(m)=$sum of all $s^i(m).$ This gives your answer.