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 group theory">group (mathematics)">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, the ring of integers. Let G be a discrete group, R a non-zero ring with a unit, and the group ring. The group G has cohomological dimension less than or equal to n, denoted, if the trivial -module R has a projective resolution of length n, i.e. there are projective -modules and -module homomorphisms and, such that the image of coincides with the kernel of for and the kernel of is trivial.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, that is, whenever. The p-cohomological dimension for prime p is similarly defined in terms of the p-torsion groups.
The smallest n such that the cohomological dimension of G is less than or equal to n 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 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.- 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.
- The fundamental group of a compact, connected, orientable Riemann surface other than the sphere has cohomological dimension two.
- 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 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 characteristic p has p-cohomological dimension at most 1.
- Every finite field has absolute Galois group isomorphic to and so has cohomological dimension 1.
- The field of formal Laurent series over an algebraically closed field k of characteristic zero also has absolute Galois group isomorphic to and so cohomological dimension 1.