Local Euler characteristic formula

Let K be a non-archimedean local field, let K^s denote a separable closure of K, let G_K = Gal(K^s/K) be the absolute Galois group of K, and let Hⁱ(K, M) denote the group cohomology of G_K with coefficients in M. Since the cohomological dimension of G_K is two,{{harvnb|Serre|2002|loc=§II.4.3}} Hⁱ(K, M) = 0 for i ≥ 3. Therefore, the Euler characteristic only involves the groups with i = 0, 1, 2.

Let M be a G_K-module of finite order m. The Euler characteristic of M is defined to beThe Euler characteristic in a cohomology theory is normally written as an alternating sum of the sizes of the cohomology groups. In this case, the alternating product is more standard.

:$\backslash chi(G\_K,M)=\backslash frac\{\backslash \#\; H^0(K,M)\backslash cdot\backslash \#\; H^2(K,M)\}\{\backslash \#\; H^1(K,M)\}$

(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{{harvnb|Milne|2006|loc=Theorem I.2.8}}

:$\backslash chi(G\_K,M)=\backslash left(\backslash \#R/mR\backslash right)^\{-1\},$

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

:$\backslash chi(G\_K,M)=p^\{-[K:\backslash mathbf\{Q\}\_p]v\_p(m)\}$

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

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

:$\backslash chi(G\_K,M)=\backslash frac\{\backslash \#\; H^0(K,M)\backslash cdot\backslash \#\; H^0(K,M^\backslash prime)\}\{\backslash \#\; H^1(K,M)\}$

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

{{Reflist}}

• {{Citation

| last=Milne

| first=James S.

| title=Arithmetic duality theorems

| url=http://jmilne.org/math/Books/adt.html

| accessdate=2010-03-27

| publisher=BookSurge, LLC

| location=Charleston, SC

| year=2006

| mr=2261462

| edition=second

| isbn=1-4196-4274-X

}}

• {{Citation | last1=Serre | first1=Jean-Pierre | author1-link= Jean-Pierre Serre | title=Galois cohomology | publisher=Springer-Verlag | location=Berlin, New York | series=Springer Monographs in Mathematics | isbn=978-3-540-42192-4 | mr=1867431 | year=2002}}, translation of Cohomologie Galoisienne, Springer-Verlag Lecture Notes 5 (1964).

Category:Algebraic number theory

Category:Galois theory