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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.

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    $\begingroup$ The reference in the thesis is not to that "Cohomologie des groupes discrets" by Serre. It's to another "Cohomologie des groupes discrets" by Serre (there seem to be at least three). $\endgroup$ Jul 15, 2021 at 18:39

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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.

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