Triangulated characterizations of singularities
Pat Lank, Sridhar Venkatesh
TL;DR
This work develops a triangulated-category framework to characterize singularities via generation. By relating the splitting of $\mathcal{O}_X \to \mathbb{R}\pi_\ast \mathcal{O}_Y$ to the level of $\mathcal{O}_X$ in the essential image of $\mathbb{R}\pi_\ast$, it unifies derived splinters, rational singularities, and Du Bois singularities under a single invariant. It provides explicit computations and bounds for one-dimensional Nagata schemes and affine cones, yielding concrete upper bounds on Rouquier dimension and on the level needed to build structure sheaves from resolutions. The results offer a practical, invariant-based approach to gauge the failure of prescribed singularity types and to quantify complexity in derived categories of singular spaces.
Abstract
This work presents a range of triangulated characterizations for important classes of singularities such as derived splinters, rational singularities, and Du Bois singularities. An invariant called 'level' in a triangulated category can be used to measure the failure of a variety to have a prescribed singularity type. We provide explicit computations of this invariant for reduced Nagata schemes of Krull dimension one and for affine cones over smooth projective hypersurfaces. Furthermore, these computations are utilized to produce upper bounds for Rouquier dimension on the respective bounded derived categories.