Local Euler characteristic formula


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 GK of a non-archimedean local field K.

Statement

Let K be a non-archimedean local field, let Ks denote a separable closure of K, let GK = Gal be the absolute Galois group of K, and let Hi denote the group cohomology of GK with coefficients in M. Since the cohomological dimension of GK is two, Hi = 0 for i ≥ 3. Therefore, the Euler characteristic only involves the groups with i = 0, 1, 2.

Case of finite modules

Let M be a GK-module of finite order m. The Euler characteristic of M is defined to be
.
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 Qp, and if vp denotes the p-adic valuation, then
where is the degree of K over Qp.
The Euler characteristic can be rewritten, using local Tate duality, as
where M′ is the local Tate dual of M.