---
tag: EF4ZWACHD8
kind: concept
name: Cohomological dimension of a group
msc: [20J05, 20J06]
sources:
  - url: https://en.wikipedia.org/wiki/Cohomological_dimension
---

## Summary

The cohomological dimension of a [[group|G5S1M9]] $G$ is the largest degree in which group cohomology of $G$ can be nonzero.
It equals the shortest length of a projective resolution of $\Z$ over the group ring $\Z[G]$.
Free groups are exactly the groups of dimension at most $1$, by the [[Stallings–Swan theorem|B41GE4M134]], and $\Z^n$ has dimension $n$.
A group of finite cohomological dimension is [[torsion-free|PEMNN32A43]].

## Explanations

### L1 human

The cohomological dimension says how many dimensions a group needs to be modelled as a shape.
The integers $\Z$ need one dimension: they are the loops on a circle.
The group $\Z^2$ needs two, like a torus.
A nontrivial finite cyclic group cannot be modelled in any finite dimension, and its dimension is infinite.