  The complexity of a module is an important homological invariant that
measures the polynomial rate of growth of its minimal projective resolution.
For the symmetric group $\Sigma_n$, the Lie module $\mathsf{Lie}(n)$ has
attracted a great deal of interest in recent years. We prove here that the
complexity of $\mathsf{Lie}(n)$ in characteristic $p$ is $t$ where $p^t$ is the
largest power of $p$ dividing $n$, thus proving a conjecture of Erdmann, Lim
and Tan. The proof uses work of Arone and Kankaanrinta which describes the
homology $\operatorname{H}_\bullet(\Sigma_n, \mathsf{Lie}(n))$ and earlier work
of Hemmer and Nakano on complexity for modules over $\Sigma_n$ that involves
restriction to Young subgroups.
