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).

[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

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