In [this thesis](http://theses.gla.ac.uk/182/) by Martin Hamilton on
Finiteness Conditions in Group Cohomology there is on page 11 a reference to following result:


**Theorem 1.2.14.** If $G$ is a *torsion-free* group and $H$ is a
subgroup of *finite index*, then


$$ \operatorname{cd} H = \operatorname{cd} G $$


where $\operatorname{cd} G $ is the cohomological dimension of $G$
defined as the **projective dimension**
of $\mathbb{Z}$ considered as $\mathbb{Z}G$-module with
trivial $G$ action, i.e. $g.1=1$ for every $g \in G$.


That is $\operatorname{cd} G = 
\operatorname{proj.dim}\_{\mathbb{Z}G} \mathbb{Z}$ and
the latter is defined as the minimal length of all projecive
resolutions


$$ 0 \to P\_n \to P\_{n-1} \to ... \to P\_1 \to P\_0 \to \mathbb{Z} \to 0 $$


of projective $\mathbb{Z}G$-modules $P\_j$.


In the thesis the author gave as reference
Jean-Pierre Serre's publication "Cohomologie des groupes discrets", can be found in this Bourbaki collection band:
<https://www.springer.com/gp/book/9783540057208>


Unfortunatelly, this result cannot be found in this publication. So my concern
is where I can find a complete proof of the quoted Theorem above.



Cohomological dimension of torsion-free groups and its subgroups
This is Theorem 3.1, p. 190, in Brown, "Cohomology of groups". He also attributes it to Serre.


As a remark, this is the reason that the virtual cohomological dimension (vcd) is well-defined.


