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 cdR(G) ≤ n, if the trivial RG-module R has a projective resolution of length n, i.e. there are projective RG-modules P0, …, Pn and RG-module homomorphisms dk: PkPk − 1 (k = 1, …, n) and d0: P0R, such that the image of dk coincides with the kernel of dk − 1 for k = 1, …, n and the kernel of dn 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 = cdR(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 cdZ(G) ≤ n.
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