  We study the homology of $[P_n,P_n]$, the commutator subgroup of the pure
braid group on $n$ strands, and show that $H_l([P_n,P_n])$ contains a free
abelian group of infinite rank for all $1\leq l\leq n-2$. As a consequence we
determine the cohomological dimension of $[P_n,P_n]$: for $n\geq 2$ we have
$\mathrm{cd}([P_n,P_n])=n-2$.
