Cohomological dimension

In abstract algebra, cohomological dimension is an invariant of a group which measures the homological complexity of its representations. It has important applications in geometric group theory, topology, and algebraic number theory.

As most (co)homological invariants, the cohomological dimension involves a choice of a "ring of coefficients" R, with a prominent special case given by R = Z, the ring of integers. Let G be a discrete group, R a non-zero ring with a unit, and RG the group ring. The group G has cohomological dimension less than or equal to n, denoted cd_R(G) ≤ n, if the trivial RG-module R has a projective resolution of length n, i.e. there are projective RG-modules P₀, …, P_n and RG-module homomorphisms d_k: P_k${\displaystyle \to }$P_k − 1 (k = 1, …, n) and d₀: P₀${\displaystyle \to }$R, such that the image of d_k coincides with the kernel of d_k − 1 for k = 1, …, n and the kernel of d_n is trivial.

...
Wikipedia