In the mathematical field of Galois cohomology, the local Euler characteristic formula is a result due to John Tate that computes the Euler characteristic of the group cohomology of the absolute Galois group G<sub>K</sub> of a non-archimedean local field K.
Let K be a non-archimedean local field, let K<sup>s</sup> denote a separable closure of K, let G<sub>K</sub> = Gal(K<sup>s</sup>/K) be the absolute Galois group of K, and let H<sup>i</sup>(K, M) denote the group cohomology of G<sub>K</sub> with coefficients in M. Since the cohomological dimension of G<sub>K</sub> is two, H<sup>i</sup>(K, M) = 0 for i âÂÂ¥ 3. Therefore, the Euler characteristic only involves the groups with i = 0, 1, 2.
Let M be a G<sub>K</sub>-module of finite order m. The Euler characteristic of M is defined to be
(the ith cohomology groups for i âÂÂ¥ 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
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<sub>p</sub>, and if v<sub>p</sub> denotes the p-adic valuation, then
where [K:Q<sub>p</sub>] is the degree of K over Q<sub>p</sub>.
The Euler characteristic can be rewritten, using local Tate duality, as
where M<sup>′</sup> is the local Tate dual of M.