# 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 (co)homological 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}

*P*

_{k − 1}(

*k*= 1, …,

*n*) and

*d*

_{0}:

*P*

_{0}

*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 coeffients 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*}.^{[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 *n* = cd_{R}(*G*).

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 cd_{Z}(*G*) ≤ *n*.

## Examples

In the first group of examples, let the ring *R* of coefficients be **Z**.

- A free group has cohomological dimension one. As shown by John Stallings (for finitely generated group) and Richard Swan (in full generality), this property characterizes free groups.
- 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 compact, 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
**Z**. More generally, the same is true for groups with nontrivial torsion.

Now let us consider the case of a general ring *R*.

- 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 (or, equivalently, the orders of its elements) is invertible in*R*. - Generalizing the Stallings–Swan theorem for
*R*=**Z**, 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*.^{[2]} The cohomological dimension of *K* is the supremum of the *p*-cohomological dimension over all primes *p*.^{[3]}

## Examples

- Every field of non-zero characteristic has cohomological dimension at most 1.
^{[4]} - Every finite field has absolute Galois group isomorphic to and so has cohomological dimension 1.
^{[5]} - The field of formal Laurent series
*k*((*t*)) over an algebraically closed field*k*of non-zero characteristic also has absolute Galois group isomorphic to and so cohomological dimension 1.^{[5]}

## See also

## References

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