Relative singularity categories, Gorenstein objects and silting theory

TL;DR

This paper introduces a new framework for studying singularity categories via Gorenstein objects in triangulated categories, extending classical notions and establishing equivalences with relative singularity categories.

Contribution

It defines $ ext{ω}$-Gorenstein objects in triangulated categories and proves their stable category is triangulated and often equivalent to the relative singularity category.

Findings

$ ext{ω}$-Gorenstein objects generalize Gorenstein modules.

Stable category of $ ext{ω}$-Gorenstein objects is triangulated.

Under certain conditions, this stable category is equivalent to the relative singularity category.

Abstract

We study singularity categories through Gorenstein objects in triangulated categories and silting theory. Let ${\omega}$ be a semi-selforthogonal (or presilting) subcategory of a triangulated category $\mathcal{T}$. We introduce the notion of $\omega$-Gorenstein objects, which is far extended version of Gorenstein projective modules and Gorenstein injective modules in triangulated categories. We prove that the stable category $\underline{\mathcal{G}_{\omega}}$, where $\mathcal{G}_{\omega}$ is the subcategory of all ${\omega}$-Gorenstein objects, is a triangulated category and it is, under some conditions, triangle equivalent to the relative singularity category of $\mathcal{T}$ with respect to $\omega$.