Local Euler characteristic formula explained

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, 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^[1]

\chi(G

K,M)=	\#H^{0(K,M) ⋅ \#}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

	-1
\chi(G
	K,M)=\left(\#R/mR\right)

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

	-[K:Q_p]v_p(m)
\chi(G
	K,M)=p

where [''K'':'''Q'''<sub>''p''</sub>] is the degree of K over Q_p.

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

\chi(G

K,M)=	\#H^{0(K,M) ⋅ \#}H^0(K,M^\prime)
	\#H^1(K,M)

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

References

- , translation of Cohomologie Galoisienne, Springer-Verlag Lecture Notes 5 (1964).

Notes and References

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.