▸ This work is concerned with approximability (via Neeman) and Rouquier dimension for triangulated categories associated to noncommutative algebras over schemes. Amongst other things, we establish that the category of perfect complexes of a coherent algebra over a separated Noetherian scheme is strongly generated if, and only if, there exists an affine open cover where the algebra has finite global dimension. As a consequence, we solve an open problem posed by Neeman. Further, as a first application, we study the existence of generators and behaviour under the derived pushforward for Azumaya algebras.

▸ We give an alternative proof of the theorem by Kuznetsov and Lunts, stating that any separated scheme of finite type over a field of characteristic zero admits a categorical resolution of singularities. Their construction makes use of the fact that every variety (over a field of characteristic zero) can be resolved by a finite sequence of blow-ups along smooth centres. We merely require the existence of (projective) resolutions. To accomplish this we put the $\mathcal{A}$-spaces of Kuznetsov and Lunts in a different light, viewing them instead as schemes endowed with finite filtrations. The categorical resolution is then constructed by gluing together differential graded categories obtained from a hypercube of finite length filtered schemes.

▸ We compute the stringy $E$-function of the affine cone over a Grassmannian. If the Grassmannian is not a projective space then its cone does not admit a crepant resolution. Nonetheless the stringy $E$-function is sometimes a polynomial and in those cases the cone admits a noncommutative crepant resolution. This raises the question as to whether the existence of a noncommutative crepant resolution implies that the stringy $E$-function is a polynomial.