The cohomological dimension of a group $G$, denoted $\mathrm{cd}(G)$, is defined to be the projective dimension (i.e. length of shortest projective resolution) of the trivial module $\mathbb{Z}\in\mathbb{Z}G$-Mod, where $\mathbb{Z}G$ is the integral group ring.
Now suppose that $\mathrm{cd}(G)<\infty$ and let $M\in\mathbb{Z}G$-Mod.
Is it always true that $\mathrm{proj.dim}_{\mathbb{Z}G}(M)\le\mathrm{cd}(G)$?
This is not true even if $G$ is the trivial group (so a $\mathbb{Z}G$-module is just an abelian group), as then $\text{cd}(G)=0$, but $\text{proj.dim.}(\mathbb{Z}/2\mathbb{Z})=1>\text{cd}(G)$.
However, it is true if $M$ is free as a $\mathbb{Z}$-module, as then if $$0\to P_n\to\cdots\to P_1\to P_0\to\mathbb{Z}\to0$$ is a projective resolution of $\mathbb{Z}$ as a $\mathbb{Z}G$-module, then $$0\to P_n\otimes_\mathbb{Z}M\to \cdots\to P_1\otimes_\mathbb{Z}M\to P_0\otimes_\mathbb{Z}M\to M\to0$$ is a projective resolution of $M$ as a $\mathbb{Z}G$-module.