Cohomological Dimension - Cohomological Dimension of A Group

Cohomological Dimension of A Group

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_kP_{k − 1} (k = 1, …, n) and d₀: P₀R, 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.

Equivalently, the cohomological dimension is less than or equal to n if for an arbitrary RG-module M, the cohomology of G with coeffients in M vanishes in degrees k > n, that is, Hk(G,M) = 0 whenever k > n.

The smallest n such that the cohomological dimension of G is less than or equal to n is the cohomological dimension of G (with coefficients R), which is denoted n = cd_R(G).

A free resolution of Z can be obtained from a free action of the group G on a contractible topological space X. In particular, if X is a contractible CW complex of dimension n with a free action of a discrete group G that permutes the cells, then cd_Z(G) ≤ n.