View source for Local Euler characteristic formula

In the [[mathematics|mathematical]] field of [[Galois cohomology]], the '''local Euler characteristic formula''' is a result due to [[John Tate (mathematician)|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''.

==Statement==
Let ''K'' be a non-archimedean local field, let ''K<sup>s</sup>'' denote a [[separable closure]] of ''K'', let ''G<sub>K</sub>''&nbsp;=&nbsp;Gal(''K<sup>s</sup>''/''K'') be the absolute Galois group of ''K'', and let ''H<sup>i</sup>''(''K'',&nbsp;''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,<ref>{{harvnb|Serre|2002|loc=§II.4.3}}</ref> ''H<sup>i</sup>''(''K'',&nbsp;''M'')&nbsp;=&nbsp;0 for ''i''&nbsp;≥&nbsp;3. Therefore, the Euler characteristic only involves the groups with ''i''&nbsp;=&nbsp;0,&nbsp;1,&nbsp;2.

===Case of finite modules===
Let ''M'' be a [[Galois module|''G<sub>K</sub>''-module]] of finite [[order (group theory)|order]] ''m''. The Euler characteristic of ''M'' is defined to be<ref>The 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.</ref>
:<math>\chi(G_K,M)=\frac{\# H^0(K,M)\cdot\# H^2(K,M)}{\# H^1(K,M)}</math>
(the ''i''th cohomology groups for ''i''&nbsp;≥&nbsp;3 appear tacitly as their sizes are all one).

Let ''R'' denote the [[local field|ring of integers]] of ''K''. Tate's result then states that if ''m'' is [[relatively prime]] to the [[characteristic (algebra)|characteristic]] of ''K'', then<ref>{{harvnb|Milne|2006|loc=Theorem I.2.8}}</ref>
:<math>\chi(G_K,M)=\left(\#R/mR\right)^{-1},</math>
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 [[local field|residue field]] of ''K'', then the Euler characteristic is one. If ''K'' is a [[finite extension]] of the [[p-adic number|''p''-adic numbers]] '''Q'''<sub>''p''</sub>, and if ''v<sub>p</sub>'' denotes the [[p-adic valuation|''p''-adic valuation]], then
:<math>\chi(G_K,M)=p^{-[K:\mathbf{Q}_p]v_p(m)}</math>
where [''K'':'''Q'''<sub>''p''</sub>] is the [[degree of a field extension|degree]] of ''K'' over '''Q'''<sub>''p''</sub>.

The Euler characteristic can be rewritten, using [[local Tate duality]], as
:<math>\chi(G_K,M)=\frac{\# H^0(K,M)\cdot\# H^0(K,M^\prime)}{\# H^1(K,M)}</math>
where ''M''<sup>&prime;</sup> is the [[local Tate dual]] of ''M''.

==Notes==
{{Reflist}}

==References==
* {{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]]