The singularity category of an exact category applied to characterize Gorenstein schemes

TL;DR

This paper extends the concept of singularity categories from Gorenstein local rings to schemes, providing new characterizations and generalizations of classic equivalences in algebraic geometry.

Contribution

It introduces a non-affine analogue of the singularity category for Gorenstein schemes and generalizes Buchweitz's equivalence to the scheme setting.

Findings

Identified a non-affine singularity category for Gorenstein schemes

Generalized Buchweitz's equivalence to schemes

Characterized rings with finite finitistic dimension

Abstract

We study singularity categories of exact categories with a focus on those associated to a complete hereditary cotorsion pair. As an application we identify a non-affine analogue of the singularity category of a Gorenstein local ring; with this Buchweitz's classic equivalence of three categories over Gorenstein local rings has been generalized to schemes, a project started by Murfet and Salarian more than ten years ago. As another application we use the framework to characterize rings of finite finitistic dimension.