Galois cohomology

Remarks

The group $H^0(K, M)$ is the group $M^{G_K}$ of fixed elements. The group $H^n(K, M)$ is also the direct limit, over finite Galois extensions $L/K$ inside $K^s$, of the group cohomology groups $H^n(\Gal(L/K), M^{\Gal(K^s/L)})$. Example: $M = (K^s)^\times$ is a discrete $G_K$-module, since each element of $K^s$ lies in a finite extension of $K$, and its stabilizer contains the open subgroup fixing that extension. For it, $H^0(K, (K^s)^\times) = K^\times$, $H^1(K, (K^s)^\times) = 0$ by Hilbert's Theorem 90, and $H^2(K, (K^s)^\times)$ is the Brauer group of $K$. For an integer $m$ not divisible by the characteristic of $K$, with $\mu_m$ the group of $m$-th roots of unity in $K^s$, Kummer theory gives $H^1(K, \mu_m) \cong K^\times / (K^\times)^m$. Non-example: let $\ell$ be a prime and $\Z_\ell(1)$ the inverse limit of the groups $\mu_{\ell^n}$ in $\overline{\Q}$, with $G_\Q$ acting on each $\mu_{\ell^n}$. With the discrete topology, $\Z_\ell(1)$ is not a discrete $G_\Q$-module. The stabilizer of a generator is the subgroup fixing all $\ell$-power roots of unity. It is closed but not open, because the field of all $\ell$-power roots of unity has infinite degree over $\Q$. For such modules one uses continuous cochains for the $\ell$-adic topology instead.