Let $k$ be a number field. Let $M$ be a (continuous) $\text{Gal}(\overline{k}/k)$-module.

One can define two subgroups of the Galois cohomology group $H^i(k,M)$:

• the group of elements of $H^i(k,M)$ mapping to zero in $H^i(\langle g \rangle, M)$ for all $g \in \text{Gal}(\overline{k}/k)$,

• the group of elements of $H^i(k,M)$ mapping to zero in $H^i(k_v,M)$ for almost all places $v$ of $k$.

Is it true that these groups coincide? And if so, how does one prove this?

Let $k$ be a number field. Let $M$ be a (continuous) $\text{Gal}(\overline{k}/k)$-module.

One can define two subgroups of the Galois cohomology group $H^i(k,M)$:

• the group of elements of $H^i(k,M)$ mapping to zero in $H^i(\langle g \rangle, M)$ for all $g \in \text{Gal}(\overline{k}/k)$,

• the group of elements of $H^i(k,M)$ mapping to zero in $H^i(k_v,M)$ for almost all places $v$ of $k$.

Is it true that these groups coincide? (The answer should be yes.) And how does one prove this?

Let $k$ be a number field. Let $M$ be a $\text{Gal}(\overline{k}/k)$-module.

One can define two subgroups of the Galois cohomology group $H^i(k,M)$:

• the group of elements of $H^i(k,M)$ mapping to zero in $H^i(\langle g \rangle, M)$ for all $g \in \text{Gal}(\overline{k}/k)$,

• the group of elements of $H^i(k,M)$ mapping to zero in $H^i(k_v,M)$ for almost all places $v$ of $k$.

Is it true that these groups coincide? And if so, how does one prove this?

Let $k$ be a number field. Let $M$ be a (continuous) $\text{Gal}(\overline{k}/k)$-module.

One can define two subgroups of the Galois cohomology group $H^i(k,M)$:

• the group of elements of $H^i(k,M)$ mapping to zero in $H^i(\langle g \rangle, M)$ for all $g \in \text{Gal}(\overline{k}/k)$,

• the group of elements of $H^i(k,M)$ mapping to zero in $H^i(k_v,M)$ for almost all places $v$ of $k$.

Is it true that these groups coincide? And if so, how does one prove this?