Let $G$ be a discrete group and $K$ be the profinite completion of $G$. Let $C_K$ denote the category of contionuous $K$-modules and ${C_K}'$ denotes category of finite continuous $K$-modules. Now for $A \in {C_K}'$, I am trying to prove the restriction map
\begin{equation} H^q(K,A) \rightarrow H^q(G,A) \end{equation} is surjective for all $q \leq n$ if for all $x \in H^q(G,A)$, $1\leq q\leq n$, there exist an $A' \in C_K$ (not necessarily finite) such that $x$ maps to $0$ in $H^q(G,A')$.
This is an exercise in Serre's Galois Cohomology. I am able to prove the statement is valid for $n=1$ without assuming the existence of $A'$. For arbitrary $n$, my idea is using the long exact sequence of cohomology groups obtained from \begin{equation} 0 \rightarrow A \rightarrow A' \rightarrow A'/A \rightarrow 0 \end{equation} and doing induction on $n$. However since $A'$ is not necessarily finite this argument does not work. Is there any way of replacing $A'\in C_K$ with some abelian group in ${C_K}'$ ? or am I missing something ? Thanks,
Ozlem
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1$\begingroup$ I can't answer the specific question but wonder what sources you've looked at. Though this kind of question arose first in the study of Galois groups as in Serre's book, there is by now a lot of literature exploring cohomology of profinite groups in a more general setting. Have you looked for instance at the book by J.S. Wilson Profinite Groups? (London Mathematical Society Monographs, New Series 19, Oxford University Press, New York, 1998) $\endgroup$– Jim HumphreysJun 24, 2014 at 15:23
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1$\begingroup$ @JimHumphreys I was reading Serre's book when I asked this question. Unfortunately there is not much detail in that book ( actually this is given in exercises). I also looked at the book you mentioned but didn't help much. $\endgroup$– ozheidiAug 17, 2014 at 13:27
1 Answer
So I think I need to find a finite sub-$K$-module $A_{\alpha}$ of $A'$ containing $A$ such that $x \in H^q(G,A)$ maps to $0$ in $H^q(G, A_{\alpha})$.
Here is the idea : Writing $ A'\cong \varinjlim_{\beta} A_{\beta}$ where the limit is taken over all finitely generated sub-$K$-modules, we have a map \begin{equation} \phi :\varinjlim_{\beta} H^q(G,A_{\beta}) \rightarrow H^q(G,\varinjlim_{\beta}A_{\beta}) \end{equation}
Now if this map is injective, I can conclude that $x$ maps to $0$ on one of these $H^q(G,A_{\alpha})$'s and maybe taking the torsion part of $A_{\alpha}$ I may be able to find such a finite sub-$K$-module of $A'$ that kills the cohomology class $x$.
I know $\phi$ is not always an isomorphism but is it possible $\phi$ to be injective ?