Categorical resolutions of filtered schemes

TL;DR

This paper provides an alternative proof that any separated scheme of finite type over a characteristic zero field admits a categorical resolution of singularities, using filtered schemes and dg categories.

Contribution

It introduces a new perspective on $ ext{A}$-spaces as filtered schemes and constructs categorical resolutions by gluing dg categories from filtered schemes.

Findings

Proves existence of categorical resolutions for schemes over characteristic zero fields.

Uses filtered schemes to simplify the construction of resolutions.

Provides a new proof approach based on gluing dg categories.

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.