Cohomological Dimension

The largest integer n (or infinity) such that a given group or topological space has nontrivial cohomology in degree n with coefficients in some module (or system of coefficients); for groups this is often the supremum of degrees in which group cohomology with appropriate modules is nonzero.

The largest integer n (or infinity) such that cohomology with specified coefficients can be nonzero; applied to spaces, groups, or schemes, it measures the topological or algebraic complexity detectable by cohomology.