Let $G\subset GL_n$ be a linear algebraic group over $\mathbb{Q}$ and let $\Gamma\subset G\cap GL_n(\mathbb{Z})$ 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 $GL_2(\mathbb{Z}_p)$. 




  [1]: https://pdf.sciencedirectassets.com/271593/1-s2.0-S0022404912X00021/1-s2.0-S0022404911002659/main.pdf?X-Amz-Security-Token=IQoJb3JpZ2luX2VjEKb%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FwEaCXVzLWVhc3QtMSJIMEYCIQC1gSkfrKJx4bPcjzMEyW5TnpYo1JucCS3wx%2BuE9UfgVgIhAImDquyUJtEGRfmeQ1uu8RJ8UPeowDkQaLZ3Vz%2Fbnq4RKr0DCJ7%2F%2F%2F%2F%2F%2F%2F%2F%2F%2FwEQAxoMMDU5MDAzNTQ2ODY1IgyCZ0En8IwdIWo7mXEqkQPBEf9lZQRJ%2BV3Ayk5YLdV2ZdnOVZYJ2PCDZ3DKTIzKAGoJsyPWZnnsao8%2B3QJrEM3Bv3CLvUUrNmiEyl4Q8YfMjH1vUeHZzexO%2BGYpY63kaciQZlczuwh5N6fXoJXth00GAWyx2gDi1ZJvfYKEEbCSDA8aOYw6rv7xucDkXs79XO9pRnku7Z8HUCkyzQxvUap6l3gvdMT5uGx3VPLqUn%2BvaEVNnaco5%2BVkC4TsrD0g2ek81jYa%2FVyiZ%2F9EZEOdVKmqLkG78EI6ZvVpwmZB1DMWFiQ8i6vQzzQsjG3p61Nsz4M3EhDP89nL2TlaIuZT855AEY5Z5B7yru%2FoWJ2ET%2BB7vYRufB46IAGGJTn9i75yZuBXhp9G4KlED2%2BKsh67wRR7IDjgcOkySMZv1KxBVGDe1VDaGW%2BY0V0GRLb9OCyvfy4stJg62YrjOCdPaVWCH1NXPw7mm0Vvo%2F9tQyS43gkEAHUQZ7pUN%2F%2BkWoiedzGs5d1SqkFYeMIOKsgNPxfQUJMq1cIRVCUn6taccFWTzONThDCf88H6BTrqASWNEWJffo7xFVTLaOqTq6%2BDDj1P8j7xFhYyC5SQuNpcQtO7xaGXfq2D%2BNiEqo08G9NgHYa9TApja0iD3KCOdVYYJq85j0qX9SSgOehio%2FDgjJTgv%2FnF1u%2BqAQ%2BWEgZVFFFgKNQYorS8yydx0KKjMXh%2FlbM9En4bgh6cTxfPcgv3yUJbTeN9WlN75PViJaMDu1Gep9ScEXHh%2FN1ndZcbCq5jEVGk4ExVIKXaAU1opgNqguQn8G%2BtZsnSorC8xLyVVI1Oy4VSRfX4fw6Zmvz71O0PVi%2FGghKRibM9TkysiP1KNdYbVISvJisUfA%3D%3D&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Date=20200903T064234Z&X-Amz-SignedHeaders=host&X-Amz-Expires=300&X-Amz-Credential=ASIAQ3PHCVTYU3YIQNHE%2F20200903%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Signature=f4af0c5d4c9253eff1d1878db24e1ea6a128e11316ffd3b22c7470538f06028b&hash=e85c33b164079f0b8ea88b5bb4888986e051dcf498d9a140751b8966a7ba6942&host=68042c943591013ac2b2430a89b270f6af2c76d8dfd086a07176afe7c76c2c61&pii=S0022404911002659&tid=spdf-46288676-9ca7-486b-ad71-f7dec68800ad&sid=0da667d553fc83493e4b4202f7b383989db5gxrqb&type=client