 Integral cohomological dimension of a group

The integral cohomological dimension is the least length of a projective resolution of the trivial group ring module $\mathbb{Z}$, equivalently the largest degree in which group cohomology with some module can be nonzero. A group containing torsion has infinite integral cohomological dimension. For ${\mathbb{Z}}^r$ the dimension is $r$, realized by its torus classifying space and detected by its top cohomology.