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(Ks/K) be the absolute Galois group of K, and let Hi(K, M) denote the group cohomology of GK with coefficients in M. Since the cohomological dimension of GK is two, Hi(K, M) = 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
χ ( G K , M ) = # H 0 ( K , M ) ⋅ # H 2 ( K , M ) # H 1 ( K , M ) {\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
χ ( G K , M ) = ( # R / m R ) − 1 , {\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 Qp, and if vp denotes the p-adic valuation, then
χ ( G K , M ) = p − [ K : Q p ] v p ( m ) {\displaystyle \chi (G_{K},M)=p^{-[K:\mathbf {Q} _{p}]v_{p}(m)}}
where [K:Qp] is the degree of K over Qp. The Euler characteristic can be rewritten, using local Tate duality, as
χ ( G K , M ) = # H 0 ( K , M ) ⋅ # H 0 ( K , M ′ ) # H 1 ( K , M ) {\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.
Notes
References Milne, James S. (2006), Arithmetic duality theorems (second ed.), Charleston, SC: BookSurge, LLC, ISBN 1-4196-4274-X, MR 2261462, retrieved 2010-03-27 Serre, Jean-Pierre (2002), Galois cohomology, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-42192-4, MR 1867431, translation of Cohomologie Galoisienne, Springer-Verlag Lecture Notes 5 (1964).
