Let $G$ be a profinite group and $M$ be a finite $G$-module. I can compute the cohomology of $G$ with coefficients in $M$ either as a topological group or as a discrete group. There is an obvious map $H^p(G,M)\to H^p(G^{\delta},M)$ (where $G^\delta$ denotes the underlying discrete group of $G$) which forgets that a $p$-cochain is continuous.
I would like to know if there are conditions on $G$ that insure that these maps are isomorphisms. In my case $G$ is finitely generated (as a topological group).
Best Answer
If $G$ is finitely generated, then $G$ is isomorphic to its profinite completion by a result of Nikolov and Segal mentioned by Ian Agol. Thus, what are you asking is equivalent to the goodness introduced by Serre (see J. P. Serre, Galois cohomology, I.2.6).
The only good finitely generated profinite groups that I know are virually polycyclic (it can be proved in the same way as Theorem 2.10 in http://arxiv.org/pdf/math/0701737.pdf). In the same paper you can find many related results. For example, a non-abelian free pro-$p$ group is not good (this is a result of A.K. Bousfield, On the p-adic completions of non-nilpotent spaces, Trans. Amer. Math. Soc. 331 (1992), 335–359).