For example, the profinite group cohomology $H^2(\hat{\mathbb Z}, \mathbb Z_p)$, where $\mathbb Z_p$ is considered as a trivial discrete $\hat{\mathbb Z}$-module, is isomorphic to $H^1(\hat{\mathbb Z},\mathbb Q_p/\mathbb Z_p)$ , (since $H^i(\hat{\mathbb Z},\mathbb Q_p)=0$ for $i>0$. i>0$). Which is isomorphic to $\mathbb Q_p/\mathbb Z_p$, hence nonzero.
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For example, the profinite group cohomology $H^2(\hat{\mathbb Z}, \mathbb Z_p)$, where $\mathbb Z_p$ is considered as a discrete $\hat{\mathbb Z}$-module, is isomorphic to $H^1(\hat{\mathbb Z},\mathbb Q_p/\mathbb Z_p)$, since $H^i(\hat{\mathbb Z},\mathbb Q_p)=0$ for $i>0$. Which is isomorphic to $\mathbb Q_p/\mathbb Z_p$, hence nonzero. |
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