In this thesis 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.