As most cohomological 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 ton, denoted cdR ≤ n, if the trivial RG-module R has a projective resolution of length n, i.e. there are projectiveRG-modules P0,..., Pn and RG-module homomorphisms dk: PkPk − 1 and d0: P0R, such that the image ofdk 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 coefficients in M vanishes in degrees k > n, that is, Hk = 0 whenever k > n. The p-cohomological dimension for prime p is similarly defined in terms of the p-torsion groups Hk. The smallest n such that the cohomological dimension of G is less than or equal ton is the cohomological dimension of G, which is denoted. A free resolution of can be obtained from a free action of the group G on a contractible topological spaceX. 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.
Examples
In the first group of examples, letthe ringR of coefficients be.
A free group has cohomological dimension one. As shown by John Stallings and Richard Swan, this property characterizes free groups. This result is known as the Stallings–Swan theorem. The Stallings-Swan theorem for a group G says that G is free if and only if every extension by G with abelian kernel is split.
More generally, the fundamental group of a closed, connected, orientable aspherical manifold of dimension n has cohomological dimension n. In particular, the fundamental group of a closed orientable hyperbolic n-manifold has cohomological dimension n.
Nontrivial finite groups have infinite cohomological dimension over. More generally, the same is true for groups with nontrivial torsion.
A group G has cohomological dimension 0 if and only if its group ring RG is semisimple. Thus a finite group has cohomological dimension 0 if and only if its order is invertible in R.
Generalizing the Stallings–Swan theorem for, Martin Dunwoody proved that a group has cohomological dimension at most one over an arbitrary ring R if and only if it is the fundamental group of a connected graph of finite groups whose orders are invertible in R.
Cohomological dimension of a field
The p-cohomological dimension of a field K is the p-cohomological dimension of the Galois group of a separable closure of K. The cohomological dimension of K is the supremum of the p-cohomological dimension over all primes p.
Examples
Every field of non-zero characteristicp has p-cohomological dimension at most 1.
