Non-commutative Hilbert schemes are smooth moduli spaces of framed quiver representations. For example, for a quiver with one vertex and no arrows they can be identified with Grassmannians and for a quiver of type $A_n$ they can be identified with flag varieties. The non-commutative Hilbert schemes possess cellular decompositions, giving rise to non-canonical bases of cohomology groups. In this talk I will discuss another basis of cohomology groups, consisting of products of Chern classes of universal vector bundles over nc-Hilbert schemes. This basis is obtained by interpreting cohomology of nc-Hilbert schemes as a module over the cohomological Hall algebra of the quiver.