Categorical resolution of singularities

TL;DR

This paper introduces the concept of a categorical resolution of singularities using smooth derived categories, providing a new framework for resolving singularities in algebraic geometry.

Contribution

It defines the notion of a smooth derived category and establishes a canonical categorical resolution for derived categories of schemes under certain conditions.

Findings

Existence of a canonical categorical resolution for $D(X)$ when the base field is perfect.

The approach applies to separated schemes of finite type with a dualizing complex.

Provides a new perspective on resolving singularities via derived categories.

Abstract

Building on the concept of a smooth DG algebra we define the notion of a smooth derived category. We the propose the definition of a categorical resolution of singularities. Our main example is the derived category $D(X)$ of quasi-coherent sheaves on a scheme $X$. We prove that $D(X)$ has a canonical categorical resolution if the base field is perfect and $X$ is a separated scheme of finite type with a dualizing complex.