Categorical resolutions of filtered schemes
Timothy De Deyn
TL;DR
This work delivers an alternative proof that every separated finite type scheme over a characteristic-zero field admits a categorical resolution by a smooth, strongly geometric triangulated category, avoiding the need for the strong form of Hironaka. The authors reframe Kuznetsov–Lunts’ A-spaces as finite-length filtered schemes and build the resolution by gluing a punctured hypercube of dg categories, ensuring acyclicity to obtain quasi-fully-faithful inclusions. The construction hinges on a robust theory of directed dg categories, hypercube totalisation, and a filtered extension of classical tools such as Rees algebras, Auslander algebras, and refined (filtered) blow-ups, culminating in a dg-enhanced categorical resolution and its 2-categorical enhancement. This framework opens the way to geometric categorical resolutions of orders over schemes and provides a systematic, functorial, and computable approach to resolving singularities at the categorical level with controlled semi-orthogonal decompositions. The methods have potential impact for noncommutative or filtered-structure contexts where traditional resolutions are unavailable or insufficient.
Abstract
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.