Tate cohomology of a cyclic group (${\hat{H}}^0(G,M),{\hat{H}}^{-1}(G,M)$)

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 $D(x)=\sigma (x)/x$.

When the two indicated groups are finite, their size ratio is multiplicative on short exact sequences. It is one for finite modules and therefore unchanged by finite-index modifications. A trivial integral module $\mathbb{Z}$ has quotient $|G|$, while an induced regular lattice ${\mathbb{Z}}_p[G]$ has both Tate groups zero and quotient one. These facts permit norm-index calculations without explicitly finding every norm.

For a cyclic extension of p-adic fields, sufficiently deep principal units are equivariantly isomorphic to an additive lattice by the p-adic logarithm. The normal basis theorem makes its Herbrand quotient one. Passing across finite unit quotients preserves it. The valuation exact sequence with quotient $\mathbb{Z}$ therefore gives the displayed formula.