Tate cohomology of a cyclic group
= Tate cohomology of a cyclic group
{c}
{title2=$\widehat H^0(G,M),\ \widehat H^{-1}(G,M)$}
For $G=\langle\sigma\rangle$, 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 $D(x)=\sigma(x)/x$.