Let $G$ be a profinite group, and $A$ a G-module. If $G$ is the projective limit of {$G_{\alpha}$}, and $A$ the direct limit of {$A_{\alpha}$}, then
$H^*(G,A)$ is isomorphic to $dir lim_{\alpha} H^*(G_{\alpha},A_{\alpha})$.
Here the cohomology groups are defined via the group of continuous functions from $G$ to $A$. At page 26 of this book it is asserted that this theorem is false on discontinuous cochains. And my question is: is there an example illustrating this statement?
Thanks in advance.
When is the cohomology of the limit not the limit of the cohomology?
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group-cohomology
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0I mean this isomorphism, which I referred to earlier as the equality: $H^*_{\text{cont.}}(G,A) \cong dir limH^*_{\text{cont.}}(G_{\alpha},A_{\alpha})$ That is to say, the "theorem" alludes to that isomorphism exactly. – 2012-12-05
1 Answers
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If you take a prime $p$ and a non-commutative finitely generated free pro-$p$ group $F$, then the second continuous cohomology group with coefficients in $\mathbb Z/p$ is trivial $H^2_{\sf cont}(F,\mathbb Z/p) =0$ but the discrete (discontinuous) cohomology group is uncountable (in particular, it is nontrivial) $H^2_{\sf disc}(F,\mathbb Z/p)\ne 0.$ (see this or this ).
Since $H^2_{\sf cont}(F,\mathbb Z/p)=\varinjlim H^2_{\sf disc}(F/U,\mathbb Z/p)$ and $F=\varprojlim F/U,$ where $U$ runs over all open normal subgroups, this is a counterexample to your statement.