= Integral cohomological dimension of a group
{title2=$\operatorname{cd}_{\mathbb Z}G$}
= Cohomological dimension
{synonym}
= Integral cohomological dimension
{synonym}
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.
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