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 G_K of a non-archimedean local field K.

Statement

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,[1] 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 G_K-module of finite order m. The Euler characteristic of M is defined to be[2]

\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[3]

\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 Q_p, and if v_p denotes the p-adic valuation, then

\chi (G_{K},M)=p^{-[K:\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

\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

Serre 2002, §II.4.3
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.
Milne 2006, Theorem I.2.8

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).

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[1] Serre 2002, §II.4.3

[2] 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.

[3] Milne 2006, Theorem I.2.8