Relative Singularity Categories and Gorenstein-Projective Modules

TL;DR

This paper develops a new framework for relative singularity categories based on self-orthogonal subcategories, linking Gorenstein-projective modules with singularity categories and establishing their properties over Gorenstein rings.

Contribution

It introduces the concept of relative singularity categories and relates them to Gorenstein-projective modules, expanding the understanding of their structure and properties.

Findings

Stable category of Gorenstein-projective modules is compactly generated.

Finitely-generated Gorenstein-projective modules form the compact objects.

Relative singularity categories are equivalent to stable categories of Cohen-Macaulay objects.

Abstract

We introduce the notion of relative singularity category with respect to any self-orthogonal subcategory $\omega$ of an abelian category. We introduce the Frobenius category of $\omega$-Cohen-Macaulay objects, and under some reasonable conditions, we show that the stable category of $\omega$-Cohen-Macaulay objects is triangle-equivalent to the relative singularity category. As applications, we relate the stable category of (unnecessarily finitely-generated) Gorenstein-projective modules with singularity categories of rings. We prove that for a Gorenstein ring, the stable category of Gorenstein-projective modules is compactly generated and its compact objects coincide with finitely-generated Gorenstein-projective modules up to direct summands.