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

Let $G$ be cyclic and let $R[G]$ carry the regular permutation action. Invariants are constant coefficient vectors, which are the images of the norm operator. A vector with coefficient sum zero is the difference of a vector and its cyclic translate, by successively solving the coordinate differences. Thus the norm group kernel equals $(\sigma -1)R[G]$. Both cyclic Tate cohomology groups vanish. This applies to $R=\mathbb{Z}$ and $R={\mathbb{Z}}_p$, and to finite direct sums of regular lattices.

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 finite module acted on by a cyclic group, write $D=\sigma -1$ and $N=1+\sigma +\cdots +{\sigma }^{n-1}$. The two Tate cohomology group sizes are $|\ker D|/|NM|$ and $|\ker N|/|DM|$. The equalities $|NM|=|M|/|\ker N|$ and $|DM|=|M|/|\ker D|$ show that these sizes are equal. The exact-sequence multiplication rule for the Herbrand quotient consequently makes it unchanged by finite-index modifications of lattices.

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.