Let $\breve{\mathbb Q}_p$ denote the completion of the maximal unramified extension of $\mathbb Q_p$. I‘d like to compute Galois cohomology groups and sets related to $\breve{\mathbb Q}_p \otimes_{\mathbb Q_p} \breve{\mathbb Q}_p =: R_2$ such as $H^i(\Gamma_{\mathbb Q_p} \times \Gamma_{\mathbb Q_p}, GL_n(R_2))$, where the $i$-th copy of $\Gamma_{\mathbb Q_p}$ (denoting the absolute Galois group of $\mathbb Q_p$) acts on the $i$-th copy of $\breve{\mathbb Q}_p$ in $R_2$.

$R_2$ seems to be an unwieldy object. Bourbaki (Algebra V.10.4. Cor.) proves that for any possibly infinite Galois extension $L/K$ there is a natural injection $L \otimes_K L \hookrightarrow \prod_{Gal(L/K)} L$. For a finite extension, this is an isomorphism of $K$-algebras. $\breve{\mathbb Q}_p$ however is only the completion of a Galois extension, so the result doesn‘t apply. Even if there is an analogous result here, it seems that it would be of limited use without a good description of the quotient.

I would be grateful for either a more workable description of $R_2$ or ideas as to how to compute its cohomology. I haven‘t had much luck considering the obvious restriction and inflation maps. For what it‘s worth, I’m interested more generally in the $n$-fold tensor product $R_n$ and in arbitrary reductive algebraic groups $G$.

Let $\breve{\mathbb Q}_p$ denote the completion of the maximal unramified extension of $\mathbb Q_p$. I‘d like to compute Galois cohomology groups and sets related to $\breve{\mathbb Q}_p \otimes_{\mathbb Q_p} \breve{\mathbb Q}_p =: R_2$ such as $H^i(\Gamma_{\mathbb Q_p} \times \Gamma_{\mathbb Q_p}, \mathrm{GL}_n(R_2))$, where the $i$-th copy of $\Gamma_{\mathbb Q_p}$ (denoting the absolute Galois group of $\mathbb Q_p$) acts on the $i$-th copy of $\breve{\mathbb Q}_p$ in $R_2$.

$R_2$ seems to be an unwieldy object. Bourbaki (Algebra V.10.4. Cor.) proves that for any possibly infinite Galois extension $L/K$ there is a natural injection $L \otimes_K L \hookrightarrow \prod_{\mathrm{Gal}(L/K)} L$. For a finite extension, this is an isomorphism of $K$-algebras. $\breve{\mathbb Q}_p$ however is only the completion of a Galois extension, so the result doesn‘t apply. Even if there is an analogous result here, it seems that it would be of limited use without a good description of the quotient.

I would be grateful for either a more workable description of $R_2$ or ideas as to how to compute its cohomology. I haven‘t had much luck considering the obvious restriction and inflation maps. For what it‘s worth, I’m interested more generally in the $n$-fold tensor product $R_n$ and in arbitrary reductive algebraic groups $G$.