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.

Cohomological dimension of a group

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 to n, denoted, if the trivial RG-module R has a projective resolution of length n, i.e. there are projective RG-modules and RG-module homomorphisms and , such that the image of d_k coincides with the kernel of d_{k-1} for 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 coefficients in M vanishes in degrees k>n, that is, whenever k>n. The p-cohomological dimension for prime p is similarly defined in terms of the p-torsion groups H^k(G,M){p}. 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. 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.

Examples

In the first group of examples, let the ring R of coefficients be \Z. Now consider the case of a general ring 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