▸ We show the first instances of schemes whose standard aisles on their derived category of quasi-coherent sheaves are not singly compactly generated.

▸ This work is concerned with categorical methods for studying singularities. Our focus is on birational derived splinters, which is a notion that extends the definition of rational singularities beyond varieties over fields of characteristic zero. Particularly, we show that an invariant called 'level' in the associated derived category measures the failure of these singularities.

▸ We prove that, given the Balmer spectrum of any essentially small monoidal-triangulated category, one has a classification of semiprime thick tensor ideals arising in terms of a "pseudo-Hochster-dual" of the noncommutative Balmer spectrum. This extends Balmer's classification of radical thick tensor ideals to noncommutative tensor-triangular geometry. To achieve this, we utilize the notion of support data for lattices and frames, under which the classification is a consequence of Stone duality. We also give a characterization for when the noncommutative Balmer spectrum behaves as it does in tensor-triangular geometry, that is, when it is a spectral space with quasi-compact opens given by complements of supports, and show that rigid centrally generated monoidal-triangulated categories satisfy this property.

▸ This work gives a new characterization of regular Noetherian schemes (more generally algebraic spaces) as those for which blowups at closed points are quasi-perfect. In particular, regularity is controlled by the property that alterations or modifications of the scheme (or algebraic space) are quasi-perfect. Additionally, we study the local behavior of quasi-perfect morphisms, showing that for proper morphisms this property can be detected via stalks, and completions and (strict) Henselizations of those. Furthermore, we show that the locus of points where a proper morphism is quasi-perfect forms a Zariski open set.

▸ Our work shows (the expected) cohomological characterization for regularity of (Noetherian) algebraic stacks; such a stack is regular if and only if all complexes with bounded and coherent cohomology are perfect. This naturally enables us to extend various statements known for schemes to algebraic stacks. In particular, the conjectures by Antieau--Gepner--Heller and Bondal--Van den Bergh, both resolved for schemes by Neeman, are proven for suitable algebraic stacks.

▸ We formalize the main approach for showing Zariski descent-type statements for strong generation of triangulated categories associated to algebro-geometric objects. This recovers various known statements in the literature. As applications we show that strong generation for the singularity category of a Noetherian separated scheme is Zariski local and obtain a strong generation result for the bounded derived category of a Noetherian concentrated algebraic stacks with finite diagonal.

▸ Our work shows forms of descent, in the fppf, h and \'{e}tale topologies, for strong generation of the bounded derived category of a noncommutative coherent algebra over a scheme. Even for (commutative) schemes this yields new perspectives. As a consequence we exhibit new examples where these bounded derived categories admit strong generators. We achieve our main results by leveraging the action of the scheme on the coherent algebra, allowing us to lift statements into the noncommutative setting. In particular, this leads to interesting applications regarding generation for Azumaya algebras.

▸ This work is concerned with approximability (à la 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 Noetherian quasi-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 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.