Purity and 2-Calabi-Yau categories

We study the Borel–Moore homology of stacks of representations of preprojective algebras ΠQ , via the study of the DT theory of the undeformed 3-Calabi–Yau completion ΠQ[x] . Via a result on the supports of the BPS sheaves for ΠQ[x]-mod , we prove purity of the BPS cohomology for the stack of ΠQ[x] -modules and define BPS sheaves for stacks of ΠQ -modules. These are mixed Hodge modules on the coarse moduli space of ΠQ -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 ΠQ -modules is also pure. We transport the cohomological wall-crossing and integrality theorems from DT theory to the category of ΠQ -modules. We use our results to prove positivity of a number of “restricted” Kac polynomials, determine the critical cohomology of Hilbn(A3) , and the Borel–Moore homology of genus one character stacks, as well as providing various applications to the cohomological Hall algebras associated to Borel–Moore homology of stacks of modules over preprojective algebras, including the PBW theorem, and torsion-freeness.