Cohomological Dimension

The integral cohomological dimension of a group is the projective dimension of the module with trivial -action over the group ring (Brown 1982, pp. 184-185). It is the shortest length of a projective resolution of that module, or if no finite length is possible. Equivalently, it is the least for which the group cohomology for every -module and every . Other coefficient rings give different versions and must be specified.

For example, a nontrivial free group has integral cohomological dimension 1. The geometric dimension of a group is at least its integral cohomological dimension. The Eilenberg-Ganea conjecture concerns equality when the latter is 2.