Equivalence of the Derived Category of a Variety with a Singularity Category

TL;DR

This paper establishes a deep connection between the derived category of a smooth variety and the singularity category of a related singular variety, extending the equivalence to the dg level.

Contribution

It proves an equivalence between the derived category of a variety and the equivariant or graded singularity category of a singular variety, including at the dg level.

Findings

Derived category of a variety is equivalent to the singularity category of a singular variety.

The equivalence extends to differential graded (dg) categories.

Provides a new perspective on the relationship between smooth and singular geometries.

Abstract

We prove an equivalence between the derived category of a variety and the equivariant/graded singularity category of a corresponding singular variety. The equivalence also holds at the dg level.