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 cohomological invariants, the cohomological dimension involves a choice of a "ring of coefficients" R, with a prominent special case given by
, the ring of integers.
Let G be a discrete group, R a non-zero ring with a unit, and
{\displaystyle RG}
the group ring.
The group G has cohomological dimension less than or equal to n, denoted
-module R has a projective resolution of length n, i.e. there are projective
-module homomorphisms
coincides with the kernel of
Equivalently, the cohomological dimension is less than or equal to n if for an arbitrary
-module M, the cohomology of G with coefficients in M vanishes in degrees
The p-cohomological dimension for prime p is similarly defined in terms of the p-torsion groups
[1] 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
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
In the first group of examples, let the ring R of coefficients be
Now consider the case of a general ring R. The p-cohomological dimension of a field K is the p-cohomological dimension of the Galois group of a separable closure of K.[4] The cohomological dimension of K is the supremum of the p-cohomological dimension over all primes p.[5]