 Tate cohomology of a cyclic group

For $G=⟨\sigma ⟩$, set $D=\sigma -1$ and $N={\sum }_{g\in G}g$. The two periodic groups are $M^G/NM$ and $\ker N/DM$. A short exact sequence of modules gives a six-term periodic exact sequence. For multiplicative modules the norm operator is a product and $D(x)=\sigma (x)/x$.