Case of finite modules

Let M be a G_K-module of finite order m. The Euler characteristic of M is defined to be^[2]

${\displaystyle \chi (G_{K},M)={\frac {\#H^{0}(K,M)\cdot \#H^{2}(K,M)}{\#H^{1}(K,M)}}}$

(the ith cohomology groups for i ≥ 3 appear tacitly as their sizes are all one).

Let R denote the ring of integers of K. Tate's result then states that if m is relatively prime to the characteristic of K, then^[3]

${\displaystyle \chi (G_{K},M)=\left(\#R/mR\right)^{-1},}$

i.e. the inverse of the order of the quotient ring R/mR.

Two special cases worth singling out are the following. If the order of M is relatively prime to the characteristic of the residue field of K, then the Euler characteristic is one. If K is a finite extension of the p-adic numbers Q_p, and if v_p denotes the p-adic valuation, then

${\displaystyle \chi (G_{K},M)=p^{-[K:\mathbf {Q} _{p}]v_{p}(m)}}$

where [K:Q_p] is the degree of K over Q_p.

The Euler characteristic can be rewritten, using local Tate duality, as

${\displaystyle \chi (G_{K},M)={\frac {\#H^{0}(K,M)\cdot \#H^{0}(K,M^{\prime })}{\#H^{1}(K,M)}}}$

where M^′ is the local Tate dual of M.