Integral cohomological dimension of a group (${\mathrm{cd}}_{\mathbb{Z}}G$)

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.

A group has integral cohomological dimension at most one precisely when it is a free group, including the trivial free group in dimension zero. This result concerns integral coefficients; a rational dimension-one statement has different conclusions.