  We study the Borel-Moore homology of stacks of representations of
preprojective algebras $\Pi_Q$, via the study of the DT theory of the
undeformed 3-Calabi-Yau completion $\Pi_Q[x]$. Via a result on the supports of
the BPS sheaves for $\Pi_Q[x]$-mod, we prove purity of the BPS cohomology for
the stack of $\Pi_Q[x]$-modules, and define BPS sheaves for stacks of
$\Pi_Q$-modules. These are mixed Hodge modules on the coarse moduli space of
$\Pi_Q$-modules that control the Borel-Moore homology and geometric
representation theory associated to these stacks. We show that the
hypercohomology of these objects is pure, and thus that the Borel-Moore
homology of stacks of $\Pi_Q$-modules is also pure.
  We transport the cohomological wall-crossing and integrality theorems from DT
theory to the category of $\Pi_Q$-modules. Among these and other applications,
we use our results to prove positivity of a number of "restricted" Kac
polynomials, determine the critical cohomology of
$\mathrm{Hilb}_n(\mathbb{A}^3)$, and the Borel-Moore homology of genus one
character stacks, as well as various applications to the cohomological Hall
algebras associated to Borel-Moore homology of stacks of preprojective
algebras, including the PBW theorem, and torsion-freeness.
