Categorical resolutions of irrational singularities

TL;DR

This paper introduces a method to embed the derived category of any singularity over a characteristic zero field into a smooth triangulated category, providing a categorical resolution of the singularity.

Contribution

It constructs categorical resolutions for irrational singularities by embedding their derived categories into smooth triangulated categories with semiorthogonal decompositions.

Findings

Derived categories of singularities can be embedded into smooth categories.

Categorical resolutions exist for all singularities over characteristic zero fields.

Semiorthogonal decompositions facilitate the resolution process.

Abstract

We show that the derived category of any singularity over a field of characteristic 0 can be embedded fully and faithfully into a smooth triangulated category which has a semiorthogonal decomposition with components equivalent to derived categories of smooth varieties. This provides a categorical resolution of the singularity.