I'll explain how certain maps from Artin stacks $\mathcal{X}$ to their coarse moduli spaces behave as though they are proper. In particular, they endow $\mathrm{H}_c(\mathcal{X},\mathcal{F})$ with a perverse filtration, for certain sheaves $\mathcal{F}$. In applications, this provides a means to endow cohomological Hall algebras with filtrations respecting the product and coproduct, with cocommutative associated graded object. The integrality conjecture from cohomological DT theory is a consequence of the PBW theorem for the resulting quantum enveloping algebra. I'll explain how this works, keeping the focus on the class of examples coming from taking the cohomology of the stack of representations of a preprojective algebra.