Cohomological dimension

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 cd"R"("G") ≤ "n", if the trivial "RG"-module "R" has a projective resolution of length "n", i.e. there are projective "RG"-modules "P"0, ..., "P""n" and "RG"-module homomorphisms "d""k": "P""k"formula_1"P""k" − 1 ("k" = 1, ..., "n") and "d"0: "P"0formula_1"R", 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 coefficients in "M" vanishes in degrees "k" > "n", that is, "H""k"("G","M") = 0 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 formula_3.
A free resolution of formula_4 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 formula_5.
Examples.
In the first group of examples, let the ring "R" of coefficients be formula_4.
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".