Suppose I have a global field $K$ and a finite Galois extension $L/K$ of Galois group $G$. It is often written without proofs (it seems that this is a very common statement) that for every $G$-module $M$ of permutation (that is a free $\mathbf{Z}$-module that admits a basis permutated by the action of $G$), the Tate-Shafarevich group of degree $2$ $$ \mathrm{Sha}^2(G,M)=\mathrm{Ker}\left(H^2(G,M)\longrightarrow\prod_{v\in\Omega_K}H^2(G_v,M)\right) $$ where $\Omega_K$ is the set of places of $K$ and $G_v=\mathrm{Gal}(L_w/K_v)$ for $w$ a place of $L$ that extends $v$, is trivial. Do you have any reference for this ? or a proof (if it's not too long) ?

Suppose I have a global field $K$ and a finite Galois extension $L/K$ of Galois group $G$. It is often written without proofs (it seems that this is a very common statement) that for every $G$-module $M$ of permutation (that is a free $\mathbf{Z}$-module that admits a basis permuted by the action of $G$), the Tate-Shafarevich group of degree $2$ $$ \mathrm{Sha}^2(G,M)=\mathrm{Ker}\left(H^2(G,M)\longrightarrow\prod_{v\in\Omega_K}H^2(G_v,M)\right) $$ where $\Omega_K$ is the set of places of $K$ and $G_v=\mathrm{Gal}(L_w/K_v)$ for $w$ a place of $L$ that extends $v$, is trivial. Do you have any reference for this ? or a proof (if it's not too long) ?

Suppose I have a global field $K$ and a finite Galois extension $L/K$ of Galois group $G$. It is often written without proofs (it seems that this is a very common statement) that for every $G$-module $M$ of permutation (that is a free $\mathbf{Z}$-module that admits a basis permutated by the action of $G$), the Tate-Shafarevich group of order $2$ $$ \mathrm{Sha}^2(G,M)=\mathrm{Ker}\left(H^2(G,M)\longrightarrow\prod_{v\in\Omega_K}H^2(G_v,M)\right) $$ where $\Omega_K$ is the set of places of $K$ and $G_v=\mathrm{Gal}(L_w/K_v)$ for $w$ a place of $L$ that extends $v$, is trivial. Do you have any reference for this ? or a proof (if it's not too long) ?

Suppose I have a global field $K$ and a finite Galois extension $L/K$ of Galois group $G$. It is often written without proofs (it seems that this is a very common statement) that for every $G$-module $M$ of permutation (that is a free $\mathbf{Z}$-module that admits a basis permutated by the action of $G$), the Tate-Shafarevich group of degree $2$ $$ \mathrm{Sha}^2(G,M)=\mathrm{Ker}\left(H^2(G,M)\longrightarrow\prod_{v\in\Omega_K}H^2(G_v,M)\right) $$ where $\Omega_K$ is the set of places of $K$ and $G_v=\mathrm{Gal}(L_w/K_v)$ for $w$ a place of $L$ that extends $v$, is trivial. Do you have any reference for this ? or a proof (if it's not too long) ?