Let $G\subset \operatorname{GL}_n$ be a linear algebraic group over $\mathbb{Q}$ and let $\Gamma\subset G\cap \operatorname{GL}_n(\mathbb{Z})$ be an arithmeric subgroup without torsion. Using a result of Borel-Serre, one shows that $\mathbb{Z}$ has a bounded resolution with finite free terms as  $\mathbb{Z}[\Gamma]$-module (see for example [section 5.1][1]). This implies that $\Gamma$ has finite cohomological dimension. Moreover, this also shows that taking group cohomology commutes with filtered colimits.

The questions are the following:

>  Does the pro-completion $\hat{\Gamma}$ of $\Gamma$ have either finite
> cohomological dimension or the compatibility with filtered colimits?
> 
> Assume that $\Gamma$ is a congruence  subgroup. Does the ''congruence'' pro-completion (i.e. the completion with respect to congruence subgroups of finite index) have finite cohomological dimension or compatibility with filtered colimits? 




In particular, I would like to know if filtered colimits commute with continuous cohomology of the Iwahori subgroup of $\operatorname{GL}_2(\mathbb{Z}_p)$. 




  [1]: https://doi.org/10.1016/j.jpaa.2011.12.001