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We study the Borel-Moore homology of stacks of representations ofpreprojective algebras $\Pi_Q$, via the study of the DT theory of theundeformed 3-Calabi-Yau completion $\Pi_Q[x]$. Via a result on the supports ofthe BPS sheaves for $\Pi_Q[x]$-mod, we prove purity of the BPS cohomology forthe 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 geometricrepresentation theory associated to these stacks. We show that thehypercohomology of these objects is pure, and thus that the Borel-Moorehomology of stacks of $\Pi_Q$-modules is also pure.  We transport the cohomological wall-crossing and integrality theorems from DTtheory to the category of $\Pi_Q$-modules. Among these and other applications,we use our results to prove positivity of a number of "restricted" Kacpolynomials, determine the critical cohomology of$\mathrm{Hilb}_n(\mathbb{A}^3)$, and the Borel-Moore homology of genus onecharacter stacks, as well as various applications to the cohomological Hallalgebras associated to Borel-Moore homology of stacks of preprojectivealgebras, including the PBW theorem, and torsion-freeness.

The integrality conjecture and the cohomology of preprojective stacks